# TrustScore: Reference-Free Evaluation of LLM Response Trustworthiness

Danna Zheng1, Danyang Liu1, Mirella Lapata1, Jeff Z. Pan1,2
1 School of Informatics, University of Edinburgh, UK
2 Huawei Edinburgh Research Centre, CSI, UK

###### Abstract

Large Language Models (LLMs) have demonstrated impressive capabilities across
various domains, prompting a surge in their practical applications. However,
concerns have arisen regarding the trustworthiness of LLMs’ outputs,
particularly in closed-book question-answering tasks, where non-experts may
struggle to identify inaccuracies due to the absence of contextual or ground
truth information. This paper introduces TrustScore, a framework based on the
concept of Behavioral Consistency, which evaluates whether an LLM’s response
aligns with its intrinsic knowledge. Additionally, TrustScore can seamlessly
integrate with fact-checking methods, which assesses alignment with external
knowledge sources. The experimental results show that TrustScore achieves
strong correlations with human judgments, surpassing existing reference-free
metrics, and achieving results on par with reference-based metrics.

##  1 Introduction

Large-scale language models (LLMs) have recently been in the spotlight due to
their impressive performance in various NLP tasks, sparking enthusiasm for
potential applications (Kaddour et al., 2023; Bubeck et al., 2023). However, a
notable concern has emerged regarding the ability of LLMs to generate
plausible yet incorrect responses (Tam et al., 2022; Liu et al., 2023; Devaraj
et al., 2022), particularly challenging for users without specialized
expertise. Consequently, users are often advised to employ LLMs in scenarios
where they can confidently assess the information provided.

This concern is particularly salient in closed-book question-answering tasks
(Roberts et al., 2020; Su et al., 2023), where the absence of contextual cues
complicates the evaluation process. Unlike tasks such as reading comprehension
(Yao et al., 2023), summarization (Aharoni et al., 2023), and natural language
inference (Chen & Eger, 2023), where answers can be verified against provided
context, assessing the trustworthiness of an LLM’s response in a closed-book
setting poses a greater challenge. In such settings, LLMs rely solely on their
parametric knowledge to generate answers, making it difficult to determine
whether these responses truly align with their underlying knowledge due to the
black-box nature of LLMs. Traditional approaches like fact-checking, which
rely on evidence from external knowledge bases, may be hindered by
inaccessible or insufficient information, rendering them ineffectively.

To address these challenges, this paper introduces TrustScore, a framework
centered around the concept of Behavioral Consistency, which evaluates whether
an LLM’s response aligns with its parametric knowledge by assessing whether
the LLM makes consitent choices among its responses and other distractors.
Behavioral consistency serves as a necessary condition for ensuring an LLM’s
responses are in alignment with its parametric knowledge, enhancing the
likelihood of trustworthiness. Furthermore, TrustScore can seamlessly
integrate with fact-checking methods, offering a more holistic evaluation of
an LLM’s response. We evaluate the effectiveness of TrustScore on FLAN-T5
(Chung et al., 2022), LLaMA (Touvron et al., 2023), and GPT-3.5, showing
strong correlation with human judgments and outperforming existing reference-
free 111Reference in this paper refers to gold answers. metrics while
achieving performance comparable to reference-based metrics.

##  2 Proposed Method

Given an LLM denoted as $M$, a context-free question represented as $q$, and
the corresponding response generated by LLM $M$ for question $q$ denoted as
$a$, the objective of TrustScore is to provide a relative score to show to
what extent can the response be trusted. The resulting TrustScore falls within
the range [0, 1]. A higher score means the response is more likely to be
trusted.

Figure 1: An illustration of Behavioral Consistency Evaluator (TrustBC). LLM
responses pass through TrustBC to determine if the model selects its original
answer consistently across multiple iterations.

The key component of TrustScore is a Behavioral Consistency (TrustBC)
evaluator as illustrated in Figure 1, which assesses whether the responses
provided by the LLM are consistent with its intrinsic parametric knowledge.
This evaluator is designed to function independently, particularly useful in
situations where external knowledge bases are not accessible, or there is not
enough information available. Moreover, TrustBC can integrate with a Fact-
Checking (TrustFC) module to further assess trustworthiness using an external
knowledge base when one is available. This dual approach allows for a
comprehensive assessment of an LLM’s response, ensuring it is evaluated both
for internal consistency and external factual consistency.

###  2.1 TrustBC Stand-Alone

The Behavior Consistency Evaluator aims to determine whether the LLM $M$
consistently selects response $r$ when presented with distractors
$R=\\{r_{1},r_{2},r_{3},...\\}$. We set a maximum check limit of $n$ times,
where LLM $M$ is tasked with choosing the correct option from five choices
(including response $r$, three random distracting options, and ‘None of the
above’) in each attempt. Note that the distracting options are different
across different attempts. If LLM $M$ does not choose response $r$ in any of
the attempts, it implies that it does not consistently provide the response to
question $q$ in line with its parametric knowledge. Ideally, if we conduct an
infinite number of checks and LLM $M$ consistently selects response $r$, it
implies that response $r$ is genuinely rooted in the LLM’s underlying
knowledge. The behavior check process terminates when inconsistent behavior
occurs (LLM $M$ does not select response $r$) or when it reaches the maximum
check limit $n$. If inconsistent behavior occurs, TrustBC return 0; otherwise,
if inconsistent behavior does not occur up to the maximum check limit $n$,
TrustBC return 1.

To generate high-quality distractors, we propose a priority-based substitution
algorithm:

Words necessitating substitution are prioritized based on their
informativeness. (1) Words exclusively present in response $r$ are given
higher priority than those occurring in both response $r$ and question $q$;
(2) Entities take precedence over other words; (3) Nouns and numbers are
prioritized over verbs and adjectives, which, in turn, take precedence over
words of other parts of speech.

Substitute words are prioritized based on their sources. (1) DBpedia entities
filtered by types are top priority; (2) words filtered by embedding similarity
have the second-highest priority; (3) words randomly sampled from the
vocabulary filtered by POS tags are the lowest priority.

Details of the priority-based substitution algorithm are in Appendix A.5.1.
Examples of the generated multi-choice questions are provided in A.6.

###  2.2 Integration of TrustBC and TrustFC

When external knowledge bases are available, TrustBC can collaborate with
TrustFC for a thorough evaluation of responses. The role of TrustFC is to
verify if a response $r$ to a question $q$ aligns with facts from external
knowledge bases. This project does not seek to pioneer new fact-checking
methods. We adopt a widely accepted approach of extracting relevant evidence
from knowledge bases and assessing its support for the response $r$ through an
entailment model. TrustOV combines scores from TrustBC and TrustFC based on
criteria detailed in Appendix A.5.3. This integration is guided by several
principles: (1) Responses supported by external knowledge are deemed more
trustworthy than those contradicted by it. (2) A response insisted upon by LLM
is more trustworthy than the response not insisted upon by LLM. (3) Factual
consistency is prioritized over behavioral consistency.

##  3 Experiment Setup

##### Data Collection

We have created a composite dataset, MixedQA, for the purpose of analyzing the
correlation between TrustScore and human evaluations. It consists of 1,000
open-ended questions randomly selected from the test sets (or development sets
when test sets are unavailable) of five datasets: Natural Questions
(Kwiatkowski et al., 2019), WebQuestions (Berant et al., 2013), TriviaQA
(Joshi et al., 2017), HotpotQA (Yang et al., 2018), and PopQA (Mallen et al.,
2022). We ignore the support context in these datasets to fit the closed-book
setting. The detailed selection criteria are in the Appendix A.3.

##### LLMs

We test our TrustScore on the answer generated by FLAN-T5-XL, LLaMA-7B, and
GPT-3.5-Turbo. LLMs perform the question-answering task in a few-shot setting.
To fully exploit TrustScore, we prompt LLM to output response in long sentence
instead of short phrase. The implementation detail including the prompts and
hyper-prameters are provided in Appendix A.5.

##### Baseline Metrics

We employ the following baselines: reference-based metrics, reference-free
metrics with and without evidence retrieval. Reference-based metrics include
popular lexical matches like Exact Match, BLEU (Papineni et al., 2002),
ROUGE-L (Lin, 2004), METEOR (Banerjee & Lavie, 2005), and pretrained
evaluators such as BERTScore (Zhang et al., 2019), BARTScore (Yuan et al.,
2021), and GPTScore222GPTScore includes several backbones, and we chose
FLAN-T5-base in this work. (Fu et al., 2023). Reference-free metrics include
BARTScore and GPTScore, both with and without evidence retrieval, for a fair
comparison with our TrustScore.

##### Evaluation

Table 1: Pearson’s $r$ correlation for baseline metrics and TrustScore against
human judgments in MixedQA. NAN indicate uncomputable correlations when all
exact match scores are 0. Unless underlined, all reported correlations are
statistically significant at $p<0.01$.

Metrics | FLAN-T5 | LLaMA | GPT-3.5
---|---|---|---
Reference-based Metrics
Exact Match | 0.635 | NAN | NAN
BLEU-1 | 0.655 | 0.435 | 0.289
BLEU-4 | 0.536 | 0.376 | 0.228
ROUGE-L | 0.607 | 0.533 | 0.382
METEOR | 0.603 | 0.536 | 0.106
BERTScore | 0.573 | 0.266 | 0.087
BARTScore-ref | 0.570 | 0.474 | 0.368
GPTScore-ref | 0.610 | 0.575 | 0.453
Reference-free Metrics (w/o Evidence Retrieval)
BARTScore-src | -0.075 | -0.099 | -0.151
GPTScore-src | -0.055 | -0.081 | -0.139
TrustBC (Ours) | 0.313 | 0.133 | 0.353
Reference-free Metrics (w/ Evidence Retrieval)
BARTScore-evid | 0.070 | 0.091 | -0.013
GPTScore-evid | 0.075 | 0.186 | 0.024
TrustFC | 0.593 | 0.533 | 0.449
TrustOV (Ours) | 0.613 | 0.539 | 0.484

To validate the effectiveness of TrustScore, we calculate the correlation
between TrustScore and human judgments on the MixedQA dataset. Humans
annotators applied binary judgments to categorize the 3,000 answers generated
by LLMs as ”correct” or ”incorrect.” We initiated this with a pilot annotation
of fifty examples, achieving a strong agreement score of 0.89. Detailed
information about the annotation process is provided in Appendix A.4.

##  4 Results

Table 1 shows Pearson’s $r$ correlations between various metric scores and
human judgments for answers produced by LLMs.

##### Behavioral Consistency Analysis

In experiments where retrieved evidence is unavailable, our TrustBC
demonstrates the highest correlation with human judgments, outperforming other
reference-free metrics. Remarkably, TrustBC also outperforms some reference-
based metrics when evaluating responses from GPT-3.5. This is particularly
significant given that, in many real-world scenarios, relevant evidence is
unavailable or insufficient, making reference-based or retrieval-based metrics
inapplicable. In such cases, our TrustBC can effectively assess response
trustworthiness. Comparing the TrustBC scores across different LLMs, we
observe that the TrustBC score for LLaMA is lower than those for FLAN-T5 and
GPT3.5. This difference could be attributed to LLaMA being a base model, while
FLAN-T5 is fine-tuned with instructions and GPT-3.5 is fine-tuned with both
instructions and human feedback. These fine-tuning processes may enhance the
behavioral consistency of the latter models.

##### Integration of TrustBC and TrustFC

In experiments where retrieved evidence is available, TrustOV demonstrates
human correlations that surpass those of existing reference-free metrics and
even outperform most reference-based metrics. We note that the TrustOV score
consistently outperforms TrustFC across various LLMs. This observation
substantiates our assertion that TrustBC is not only an effective reference-
free metric on its own but can also complement TrustFC to achieve better
performance in a reference-based setting.

##### Robustness to Diverse Answers

From Table 1, we observe that traditional lexical overlapping-based metrics
(e.g., Exact Match, BLEU, and ROUGE, etc.) achieve higher correlation scores
for responses generated by FLAN-T5, but lower scores for answers generated by
GPT3.5. This discrepancy can be attributed to FLAN-T5’s tendency to produce
concise answers. Some of these answers match reference answers exactly, making
them easier to evaluate using reference-based metrics. Conversely, GPT-3.5
tends to generate longer and more diverse answers, which may deviate from
reference answers, making lexical-based metrics less effective. In contrast to
these lexical overlapping-based metrics, TrustScore excels in evaluating
GPT-3.5-generated responses and also achieves comparable results for
FLAN-T5-generated responses. This highlights its robustness across different
types of answers.

##  5 Related Work

##### QA Evaluation Metrics

Traditional metrics such as Exact Match, F1, BLEU (Papineni et al., 2002),
Rouge (Lin, 2004), and METEOR (Banerjee & Lavie, 2005), rely on lexical
matching, often overlooking semantic content. BERTScore (Zhang et al., 2019),
BARTScore (Yuan et al., 2021) and GPTScore(Fu et al., 2023) addresses this
limitation by leveraging contextualized embeddings to capture semantic
similarity. However, these methods are predominantly designed for reference-
based evaluation. While BARTScore and GPTScore can also be applied in a
reference-free manner by directly comparing generated text to source
sentences, their effectiveness is noticeably context-sensitive especially in
QA evaluation due to substantial divergence between answers and questions.

##### Behavior Consistency

Jang et al. (2022) introduced the BECEL benchmark, evaluating language models
across tasks like natural language inference and semantic analysis,
highlighting the importance of behavioral consistency. Asai & Hajishirzi
(2020) and Ribeiro et al. (2019) improved consistency in question-answering by
enriching training data with logical and linguistic insights. Cohen et al.
(2023) proposed the LM vs LM framework for factuality evaluation through
cross-examination between LMs, using multi-turn interactions to uncover
inconsistencies. However, this model relies on the subjective judgment of the
examiner LM, introducing potential bias. Our approach contrasts by offering an
objective mechanism for identifying inconsistencies without the constraints of
LM vs LM, such as needing LMs with over 10 billion parameters and reliance on
instruction-folowing and long-context reasoning ability. Additionally, unlike
LM vs LM, which depends solely on consistency for correctness, our method
integrates external knowledge for a more comprehensive evaluation.

##  6 Conclusion

We introduce TrustScore, a novel reference-free evaluation framework that
assesses the trustworthiness of an LLM response from (1) behavior consistency
of the LLM in making coherent choices among its response and other distractor,
and (2) the factuality of their generated response. Our experiments, conducted
on our newly curated MixedQA dataset, demonstrate that the two components
complement each other effectively within the TrustScore framework. TrustScore
exhibits a strong correlation with human judgments, surpassing existing
reference-free metrics, and achieves results comparable to reference-based
metrics.

## References

  * Aharoni et al. (2023) Roee Aharoni, Shashi Narayan, Joshua Maynez, Jonathan Herzig, Elizabeth Clark, and Mirella Lapata.  Multilingual summarization with factual consistency evaluation.  In _Findings of the Association for Computational Linguistics: ACL 2023_ , pp. 3562–3591, 2023.
  * Asai & Hajishirzi (2020) Akari Asai and Hannaneh Hajishirzi.  Logic-guided data augmentation and regularization for consistent question answering.  _arXiv preprint arXiv:2004.10157_ , 2020.
  * Banerjee & Lavie (2005) Satanjeev Banerjee and Alon Lavie.  Meteor: An automatic metric for mt evaluation with improved correlation with human judgments.  In _Proceedings of the acl workshop on intrinsic and extrinsic evaluation measures for machine translation and/or summarization_ , pp. 65–72, 2005.
  * Berant et al. (2013) Jonathan Berant, Andrew Chou, Roy Frostig, and Percy Liang.  Semantic parsing on freebase from question-answer pairs.  In _Proceedings of the 2013 conference on empirical methods in natural language processing_ , pp. 1533–1544, 2013.
  * Bubeck et al. (2023) Sébastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott Lundberg, et al.  Sparks of artificial general intelligence: Early experiments with gpt-4.  _arXiv preprint arXiv:2303.12712_ , 2023.
  * Chen & Eger (2023) Yanran Chen and Steffen Eger.  Menli: Robust evaluation metrics from natural language inference.  _Transactions of the Association for Computational Linguistics_ , 11:804–825, 2023.
  * Chung et al. (2022) Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Eric Li, Xuezhi Wang, Mostafa Dehghani, Siddhartha Brahma, et al.  Scaling instruction-finetuned language models.  _arXiv preprint arXiv:2210.11416_ , 2022.
  * Cohen (1960) Jacob Cohen.  A coefficient of agreement for nominal scales.  _Educational and psychological measurement_ , 20(1):37–46, 1960.
  * Cohen et al. (2023) Roi Cohen, May Hamri, Mor Geva, and Amir Globerson.  Lm vs lm: Detecting factual errors via cross examination.  _arXiv preprint arXiv:2305.13281_ , 2023.
  * Devaraj et al. (2022) Ashwin Devaraj, William Sheffield, Byron C Wallace, and Junyi Jessy Li.  Evaluating factuality in text simplification.  In _Proceedings of the conference. Association for Computational Linguistics. Meeting_ , volume 2022, pp. 7331. NIH Public Access, 2022.
  * Fu et al. (2023) Jinlan Fu, See-Kiong Ng, Zhengbao Jiang, and Pengfei Liu.  Gptscore: Evaluate as you desire.  _arXiv preprint arXiv:2302.04166_ , 2023.
  * Jang et al. (2022) Myeongjun Jang, Deuk Sin Kwon, and Thomas Lukasiewicz.  Becel: Benchmark for consistency evaluation of language models.  In _Proceedings of the 29th International Conference on Computational Linguistics_ , pp. 3680–3696, 2022.
  * Joshi et al. (2017) Mandar Joshi, Eunsol Choi, Daniel S Weld, and Luke Zettlemoyer.  Triviaqa: A large scale distantly supervised challenge dataset for reading comprehension.  _arXiv preprint arXiv:1705.03551_ , 2017.
  * Kaddour et al. (2023) Jean Kaddour, Joshua Harris, Maximilian Mozes, Herbie Bradley, Roberta Raileanu, and Robert McHardy.  Challenges and applications of large language models.  _arXiv preprint arXiv:2307.10169_ , 2023.
  * Karpukhin et al. (2020) Vladimir Karpukhin, Barlas Oguz, Sewon Min, Patrick Lewis, Ledell Wu, Sergey Edunov, Danqi Chen, and Wen-tau Yih.  Dense passage retrieval for open-domain question answering.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_. Association for Computational Linguistics, 2020.
  * Kwiatkowski et al. (2019) Tom Kwiatkowski, Jennimaria Palomaki, Olivia Redfield, Michael Collins, Ankur Parikh, Chris Alberti, Danielle Epstein, Illia Polosukhin, Jacob Devlin, Kenton Lee, et al.  Natural questions: a benchmark for question answering research.  _Transactions of the Association for Computational Linguistics_ , 7:453–466, 2019.
  * Lin (2004) Chin-Yew Lin.  Rouge: A package for automatic evaluation of summaries.  In _Text summarization branches out_ , pp. 74–81, 2004.
  * Liu et al. (2023) Yang Liu, Yuanshun Yao, Jean-Francois Ton, Xiaoying Zhang, Ruocheng Guo Hao Cheng, Yegor Klochkov, Muhammad Faaiz Taufiq, and Hang Li.  Trustworthy llms: a survey and guideline for evaluating large language models’ alignment.  _arXiv preprint arXiv:2308.05374_ , 2023.
  * Mallen et al. (2022) Alex Mallen, Akari Asai, Victor Zhong, Rajarshi Das, Hannaneh Hajishirzi, and Daniel Khashabi.  When not to trust language models: Investigating effectiveness and limitations of parametric and non-parametric memories.  _arXiv preprint arXiv:2212.10511_ , 2022.
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu.  Bleu: a method for automatic evaluation of machine translation.  In _Proceedings of the 40th annual meeting of the Association for Computational Linguistics_ , pp. 311–318, 2002.
  * Qin et al. (2023) Chengwei Qin, Aston Zhang, Zhuosheng Zhang, Jiaao Chen, Michihiro Yasunaga, and Diyi Yang.  Is chatgpt a general-purpose natural language processing task solver?  _arXiv preprint arXiv:2302.06476_ , 2023.
  * Ribeiro et al. (2019) Marco Tulio Ribeiro, Carlos Guestrin, and Sameer Singh.  Are red roses red? evaluating consistency of question-answering models.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pp. 6174–6184, 2019.
  * Roberts et al. (2020) Adam Roberts, Colin Raffel, and Noam Shazeer.  How much knowledge can you pack into the parameters of a language model?  _arXiv preprint arXiv:2002.08910_ , 2020.
  * Robertson et al. (2009) Stephen Robertson, Hugo Zaragoza, et al.  The probabilistic relevance framework: Bm25 and beyond.  _Foundations and Trends® in Information Retrieval_ , 3(4):333–389, 2009.
  * Su et al. (2023) Dan Su, Mostofa Patwary, Shrimai Prabhumoye, Peng Xu, Ryan Prenger, Mohammad Shoeybi, Pascale Fung, Animashree Anandkumar, and Bryan Catanzaro.  Context generation improves open domain question answering.  In _Findings of the Association for Computational Linguistics: EACL 2023_ , pp. 781–796, 2023.
  * Tam et al. (2022) Derek Tam, Anisha Mascarenhas, Shiyue Zhang, Sarah Kwan, Mohit Bansal, and Colin Raffel.  Evaluating the factual consistency of large language models through summarization.  _arXiv preprint arXiv:2211.08412_ , 2022.
  * Touvron et al. (2023) Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al.  Llama: Open and efficient foundation language models.  _arXiv preprint arXiv:2302.13971_ , 2023.
  * Yang et al. (2018) Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William W Cohen, Ruslan Salakhutdinov, and Christopher D Manning.  Hotpotqa: A dataset for diverse, explainable multi-hop question answering.  _arXiv preprint arXiv:1809.09600_ , 2018.
  * Yao et al. (2023) Zijun Yao, Yantao Liu, Xin Lv, Shulin Cao, Jifan Yu, Juanzi Li, and Lei Hou.  Korc: Knowledge oriented reading comprehension benchmark for deep text understanding.  In _Findings of the Association for Computational Linguistics: ACL 2023_ , pp. 11689–11707, 2023.
  * Yuan et al. (2021) Weizhe Yuan, Graham Neubig, and Pengfei Liu.  Bartscore: Evaluating generated text as text generation.  _Advances in Neural Information Processing Systems_ , 34:27263–27277, 2021.
  * Zhang et al. (2019) Tianyi Zhang, Varsha Kishore, Felix Wu, Kilian Q Weinberger, and Yoav Artzi.  Bertscore: Evaluating text generation with bert.  _arXiv preprint arXiv:1904.09675_ , 2019.

##  Appendix A Appendix

###  A.1 Limitations

Due to the high cost of collecting human evaluation as gold labels and
executing LLMs like GPT-3.5, which we include in our experiments, we do not
test TrustScore on large-scale data. Instead, we test it on MixedQA, which
consists of 1000 open-ended questions randomly collected from five question-
answering datasets.

In this work, we do not test TrustScore on long-form questions that require
elaborate and in-depth answers. We contend that our general framework can be
adapted to the long-form question-answer task. We will leave it in our future
work.

###  A.2 Ethics Statement

The MixedQA dataset is collected from several datasets that are public for
academic use and do not contain sensitive information. We will make the
dataset and the codebase of the project freely available online for academic
use.

Our human evaluation involves human participation. Two annotators were
recruited and trained. They were asked to annotate the correctness of answers
and filter out samples that might cause ethical problems. No sensitive
personal information is involved in the process.

###  A.3 Data Selection Criteria for MixedQA

We collected MixedQA from a diverse set of five datasets:

##### Natural Questions (Kwiatkowski et al., 2019)

This dataset consists of questions extracted from web queries, with each
question accompanied by a corresponding Wikipedia article containing the
answer.

##### WebQuestions (Berant et al., 2013)

This dataset comprises questions originating from web queries, meticulously
matched with entries in FreeBase.

##### TriviaQA (Joshi et al., 2017)

This dataset incorporated a collection of questions sourced from Quiz League
websites. Each question in this dataset is complemented by web pages and
Wikipedia searches that may potentially contain the answer.

##### HotpotQA (Yang et al., 2018)

This dataset primarily contains Wikipedia-based multi-hop questions, with
sentence-level supporting facts annotation.

##### PopQA (Mallen et al., 2022)

This dataset is composed of questions aimed at covering factual information in
the less frequently explored areas, essentially addressing the long tail of
information.

We extracted questions exclusively from the test or development sets,
discarding the supporting context to adapt to a closed-book setting. Our data
collection process involved the following steps:

  1. 1.

Open-Ended Questions Extraction: Initially, we utilized an automated script to
filter out comparison questions (e.g.,“Who was born earlier, Emma Bull or
Virginia Woolf?”) and “yes or no” questions (e.g., “Is Ferocactus a type of
plant?”) in order to focus on open-ended questions.

  2. 2.

Input to Language Models: We input the filtered questions, devoid of any
contextual information, to three language models: FLAN-T5-XL, LLaMA-7B, and
GPT-3.5-Turbo.

  3. 3.

Answerable Questions Extraction: We observed that FLAN-T5 occasionally
generated subpar answers, characterized by repetitive words or even blank
responses. LLaMA-7B, on the other hand, sometimes failed to provide a relevant
answer and instead repeated the original question. ChatGPT’s responses,
meanwhile, often fell into the category of inconclusive answers, with
statements like “I am not sure,” “I don’t know,” or “The information is not
provided in the question.” Such responses lacked the necessary informative
content required for trustworthiness verification. To address this issue, we
employed an automated script to filter out questions with such responses,
retaining only those questions that could be answered by the three LLMs.

  4. 4.

Random Sampling: Finally, to ensure a balanced and representative dataset, we
randomly selected 200 questions from each of the five datasets for TrustScore
evaluation.

###  A.4 Human Evaluation Details

The 1,000 questions are evenly distributed between two annotators. Each
question accompanied by answers from three LLMs: FLAN-T5, LLaMA, and GPT-3.5.
These questions were also paired with ground truth answers from the original
dataset. During our pilot study, we discovered a number of incorrect ground
truth answers. As a result, annotators were advised to use these answers as a
guide and to verify the correctness of answers via manual internet searches.

System | Cohen’s $\kappa$
---|---
FLAN-T5 | 0.88
LLaMA | 0.82
GPT | 0.92
Overall | 0.89
Table 2: Inter-annotator agreement for the double annotations on fifty
examples.

We evaluated the reliability of human annotations by measuring the agreement
between annotators. Table 2 shows the results of the Cohen’s $\kappa$ Cohen
(1960), which accounts for agreement that is expected by chance. The
annotations on answers of FLAN-T5, LLaMA, and GPT have Cohen’s $\kappa$ values
of 0.88, 0.82, and 0.92, respectively, indicating a high level of agreement.
The overall Cohen’s $\kappa$ value is 0.89, confirming the quality of the
annotations in the MixedQA dataset.

###  A.5 Implementation Details

####  A.5.1 Details of the Substitution Algorithm

To generate high-quality distractors, we propose a substitution algorithm that
produces a dictionary of substitute candidates $D$, where keys represent words
to be substituted and the values are the corresponding potential substitute
words. As shown in Algorithm1, the substitution algorithm involves three
functions to search for substitute candidates for a given entity or word:
find_dbpedia_ents, find_candidates_by_embedding, and find_candidate_by_pos.
The find_dbpedia_ents function retrieves DBpedia entities with the same
DBpedia types as the entities to be substituted. We employ the DBpedia
Spotlight for Spacy to detect DBpedia entities in response $r$ and obtain
their corresponding DBpedia types. For example, the DBpedia types for an
entity like ”Kobe” might include ”DBpedia:Person,” ”DBpedia:Athlete,” and
”DBpedia:BasketballPlayer.” Subsequently, we use SPARQL to search for DBpedia
entities that satisfy these types, starting with entities that match all types
and gradually lowering the number of required matching types if necessary. The
find_candidates_by_embedding function identifies the most similar words by
computing cosine similarity between the mean of the projection weight vectors
of the words to be substituted and the vectors for each word in a pretrained
word2vec model (word2vec-google-news-300). The find_candidate_by_pos function
randomly selects words with the same POS tag from the word2vec-google-news-300
corpus.

Once we obtain the substitute candidate dictionary $D$, we generate a set of
distractors by replacing one word or entity in response $r$ with its
candidates in each iteration. This approach minimizes the edit distance
between the distractors and response $r$, making it more challenging for LLM
$M$ to distinguish between the options.

Input: Question $q$, answer $a$, candidate minimum number $n$

Output: Substitute candidate dictionary $D$

1

2

3 Function  _Find_candidates_for_entities(_$entities\\_list$ , $num$_)_:

4 $candidate\\_dict$ = $defaultdict(list)$

5

/* search candidates for DBpedia entities (source: DBperdia) */

6 for  _each $entity$ in $entities\\_list$ _ do

7 if  _the $entity$ is a DBpedia entity_ then

8 $candidate\\_dict[entity]$ = find_dbpedia_ents$(entity,num)$

9

10

11 delete the item whose value is an empty list in $candidate\\_dict$

/* search candidates for entities (source: embedding similarity) */

12 if  _the candidate number in $candidate\\_dict$ ¡ $num$_ then

13 for  _each $entity$ in $entities\\_list$ _ do

14 $candidate\\_dict[entity]$ += find_candidates_by_embedding$(entity,num)$

15

16

17 delete the item whose value is an empty list in $candidate\\_dict$

/* search candidates for each word in entities (source: embedding similarity)
*/

18 if  _the candidate number in $candidate\\_dict$ ¡ $num$ and there exits
multi-word entity in $entities\\_list$_ then

19 for  _each $entity$ in the multi-word entities_ do

20 $candidate\\_dict[entity]$ += [replace the non-high frequency word in the
entity with the candidates searched through
find_candidates_by_embedding$(word,num)$]

21

22

23 delete the item whose value is an empty list in $candidate\\_dict$

/* search candidates for entities (source: randomly selected from corpus with
same pos) */

24 if  _the candidate number in $candidate\\_dict$ ¡ $num$_ then

25 for  _each $entity$ in $entities\\_list$ _ do

26 $candidate\\_dict[entity]$ += find_candidates_by_pos$(entity,num)$

27

28

29 return $candidate\\_dict$

30

31

32

33 Function  _Find_candidates_for_words(_$words\\_list$ , $num$_)_:

34 $candidate\\_dict$ = $defaultdict(list)$

/* search candidates for words (source: embedding similarity) */

35 for  _$word$ in $words\\_list$_ do

36 $candidate\\_dict[word]$ = find_candidates_by_embedding$(word,num)$

37

38 delete the item whose value is an empty list in $candidate\\_dict$

/* search candidates for words (source: randomly selected from corpus with
same pos) */

39 if  _the candidate number in $candidate\\_dict$ ¡ $num$_ then

40 for  _$ent$ in $words\\_list$_ do

41 $candidate\\_dict[word]$ = find_candidates_by_pos$(word,num)$

42

43 return $candidate\\_dict$

44

45

46

47 Function  _Main_ :

$ents_{1},ents_{2}$ = get entities in answer $a$

// 1 only in answer, 2 also in question

48 $nncds_{1},vbjjs_{1},nncds_{2},others$ = split words in answer $a$ with
their pos tag

49 $level_{1},level_{2},level_{3},level_{4}$ =
$ents_{1},nncds_{1},vbjjs_{1}+ents_{2}+nncds_{2},others$

50

51 $D$ = $defaultdict(list)$

52

53 if  _$level_{1}$_ then

54 $D$ = Find_candidates_for_entities$(level_{1},n)$

55 else if  _$level_{2}$_ then

56 $D$= Find_candidates_for_words$(level_{2},n)$

57 else if  _$level_{3}$_ then

58 if  _$vbjjs_{1}$_ then

59 $D$ = Find_candidates_for_words$(vbjjs_{1},n)$

60

61 if  _the candidate number in $D$ ¡ $n$ and $ents_{2}$ exits_ then

62 $D$ = merge($D$, Find_candidates_for_entities$(ents_{2},n)$)

63

64 if  _the candidate number in $D$ ¡ $n$ and $nncds_{2}$ exits_ then

65 $D$ = merge($D$, Find_candidates_for_words$(nncds_{2},n)$)

66

67 else if  _$level_{4}$_ then

68 $D$=Find_candidates_for_words$(level_{4},n)$

69 else

70 print(”I did not find the token that needs to be replaced.”)

71 return D

72

Algorithm 1 Priority-based Substitution Algorithm

####  A.5.2 TrustFC

We retrieve the top 10 related passages as evidence $e$ for the query ”answer
$ar$” correctly answers the question of ”question $q$” using a BM25-based
Robertson et al. (2009) method from the wikipedia-dpr corpus Karpukhin et al.
(2020). Following Qin et al. (2023), we employ GPT-Turbo-3.5 as an entailment
model to judge whether the evidence $e$ can validate the response $r$
generated for question $q$.

####  A.5.3 Scoring Criterion

The scoring criterion is shown in Table 3.

Table 3: The scoring criterion for TrustOV which integrate the TrustBC and TrustFC. | TrustBC
---
| TrustFC
---
TrustOV
Consistent | Support | 1
Inconsistent | Support | 0.8
Consistent | Neutral | 0.6
Inconsistent | Neutral | 0.4
Consistent | Contradict | 0.2
Inconsistent | Contradict | 0

####  A.5.4 Hyper-Parameters

FLAN-T5 and LLaMA employ a greedy decoding strategy with a maximum length of
1042 tokens. GPT-3.5 uses a temperature setting of 0 with a maximum length of
1042 tokens. The maximum check limit of behavior consistency evaluator is set
as 10.

####  A.5.5 Prompts

Table 4 presents the prompt used in closed-book question answering task.

Table 4: The prompt for Close-book Question Answering Prompt for Close-book
Question Answering
---
INSTRUCTION: Please give answers to the following questions about knowledge.
Question: who has been ranked no. 1 in the latest football rankings announced
by fifa?
Answer: Germany has been ranked no. 1 in the latest football rankings
announced by fifa.
Question: who sings i just want to use your love tonight?
Answer: English rock band the Outfield sings i just want to use your love
tonight.
Question: where was the movie the glass castle filmed?
Answer: The movie the glass castle was filmed in Welch, West Virginia.
Question: who was the first lady nominated member of the rajya sabha?
Answer: Mary Kom was the first lady nominated member of the rajya sabha.
Question: what is the tiger’s name in life of pi?
Answer: Richard Parker is the tiger’s name in life of pi.
Question: {Q}
Answer:

Table 5 presents the prompt used in behavior consistency evaluator.

Table 5: The prompt for Behavior Consistency Evaluation Prompt for Behavior
Consistency Evaluation
---
INSTRUCTION: Please give answers to the following multi-choice questions about
knowledge.
Question: who has been ranked no. 1 in the latest football rankings announced
by fifa?
A) Germany has been ranked no. 1 in the latest football rankings announced by
fifa.
B) India has been ranked no. 1 in the latest football rankings announced by
fifa.
C) Canada has been ranked no. 1 in the latest football rankings announced by
fifa.
D) Austria has been ranked no. 1 in the latest football rankings announced by
fifa.
E) None of above.
Answer: E
Question: who sings i just want to use your love tonight?
A) Latin rock band the Outfield sings i just want to use your love tonight.
B) English Power pop band the Outfield sings i just want to use your love
tonight.
C) English rock band the Outfield sings i just want to use your love tonight.
D) English melodic sensibility band the Outfield sings i just want to use your
love tonight.
E) None of above.
Answer: C
Question: where was the movie the glass castle filmed?
A) The movie the glass castle was filmed in London.
B) The movie the glass castle was filmed in Welch, West Virginia.
C) The movie the glass castle was filmed in Philadelphia.
D) The movie the glass castle was filmed in Budapest.
E) None of above.
Answer: B
Question: who was the first lady nominated member of the rajya sabha?
A) William Randolph Hearst was the first lady nominated member of the rajya
sabha.
B) Jesse Speight was the first lady nominated member of the rajya sabha.
C) Thurlow Weed was the first lady nominated member of the rajya sabha.
D) Mary Kom was the first lady nominated member of the rajya sabha.
E) None of above.
Answer: D
Question: what is on a mcchicken sandwich from mcdonalds?
A) A breaded chicken patty is on a mcchicken sandwich from mcdonalds.
B) A Hot dog chicken patty is on a mcchicken sandwich from mcdonalds.
C) A breaded Bacon is on a mcchicken sandwich from mcdonalds.
D) A breaded Teriyaki chicken is on a mcchicken sandwich from mcdonalds.
E) None of above.
Answer: A
Question: {Q}
Answer:

Table 6 presents the prompt used in factual consistency evaluator.

Table 6: Prompt for Fact Checking Prompt for Fact Checking
---
INSTRUCTION: Assess whether the provided evidence aligns with or contradicts
the given question-answer pair, and categorize the relationship as either
‘support’, ‘contradict’, or ‘neutral’.
Evidence: [evidence $e$]
Question: [question $q$]
Answer: [answer $a$]
Relation:

###  A.6 Examples of Multi-Choice Questions generated for Behavior Consistency
Evaluation

Table 7 presents the examples of the multi-choice questions used in Behavior
Consistency Evaluator

Table 7: Examples of Multi-Choice Questions Generated for Behavior Consistency
Evaluation. Examples of Multi-Choice Questions Generated for Behavior
Consistency Evaluation
---
When did all night long come out lionel richie?
A) All night long came out in 1975. [distractor]
B) All night long came out in 1986. [distractor]
C) All night long came out in 1983. [original response]
D) All night long came out in 1999. [distractor]
E) None of the above.
Who built the first temple for god in jerusalem?
A) Gopala III built the first temple for god in jerusalem. [distractor]
B) King Solomon built the first temple for god in jerusalem. [original
response]
C) Mohammad Hatta built the first temple for god in jerusalem. [distractor]
D) Alexander the Great built the first temple for god in jerusalem.
[distractor]
E) None of the above.
which state was returned to spain after the revolutionary war?
A) Illinois was returned to spain after the revolutionary war. [distractor]
B) Louisiana was returned to spain after the revolutionary war. [distractor]
C) Florida was returned to spain after the revolutionary war. [original
response]
D) Lincolnshire was returned to spain after the revolutionary war.
[distractor]
E) None of the above.


The PTA Hellings and Downs Correlation

Unmasked by Symmetries

A. Kehagias, 1 and A. Riotto,2

1 Physics Division, National Technical University of Athens, Athens, 15780,
Greece

2 Department of Theoretical Physics and Gravitational Wave Science Center
(GWSC),
Université de Genève, CH-1211 Geneva

Abstract
The Hellings and Downs correlation curve describes the correlation of the
timing residuals from pairs of pulsars as a function of their angular
separation on the sky and is a smoking-gun signature for the detection of an
isotropic stochastic background of gravitational waves. We show that it can be
easily obtained from realizing that Lorentz transformations are conformal
transformations on the celestial sphere and from the conformal properties of
the two-point correlation of the timing residuals. This result allows several
generalizations, e.g. the calculation of the three-point correlator of the
time residuals and the inclusion of additional polarization modes (vector
and/or scalar) arising in alternative theories of gravity.

###### Contents

  1. 1 Introduction
  2. 2 The Lorentz group, inertial observers and the celestial sphere
    1. 2.1 Lorentz transformations and the celestial sphere
  3. 3 The two- and three-point Correlators on the Celestial Sphere
  4. 4 Hellings and Downs from Lorentz invariance or conformal symmetry on the celestial sphere
    1. 4.1 Recovering the Hellings and Downs correlation
    2. 4.2 Going beyond the Hellings and Downs correlation: the three-point correlator
  5. 5 Recovering the overlap function for an array of “short-arm” LIGO-like detectors
  6. 6 Conclusions
  7. A The logarithmic origin of the Hellings and Downs function
  8. B The correlator for collinear pulsars
  9. C Proof that the integral of the HD correlation function over the celestial sphere vanishes

##  1 Introduction

The recent Pulsar Timing Arrays (PTA) collaborations, NANOGrav [1, 2], EPTA
[3, 4, 5], PPTA [6, 7, 8] and CPTA [9], have recently released data showing
evidence for the presence of a stochastic background of Gravitational Waves
(GWs). Pulsars are high-quality clocks generating pulses in the radio
frequencies band which arrive at Earth with regular and predictable arrival
times. The latter are slightly perturbed by the presence of the GWs which
induce correlated perturbations between different radio pulsars, called
“timing residuals”.

The correlations are predicted to follow the so-called “Hellings and Downs”
(HD) correlation curve [10] which describe the expected correlation in the
timing residuals from a pair of pulsars as a function of their angular
separation on the sky. It is a prediction of general relativity for an
unpolarized and isotropic GW background (see the recent Ref. [11] for a nice
set of comments about the HD correlation).

The goal of this paper is to show that the HD correlation function can be
obtained by very general symmetry arguments. The key point is the realization
that

  1. 1.

Lorentz transformations between inertial observers are conformal motions of
the surface of a unit two-sphere;

  2. 2.

such Lorentz transformations can be interpreted as conformal transformations
on the celestial sphere;

  3. 3.

under such transformations the HD correlation transforms in a well-defined way
with a well-defined conformal weight.

This allows one to obtain the HD correlation with no explicit and cumbersome
calculation. The symmetry arguments are also suitable to generalize the HD
correlation to the case in which three or more pulsars are correlated amd to
include additional polarization modes (vector and/or scalar) in theories of
gravity different from general relativity.

The paper is organized as follows. In section 2 we describe the Lorentz group
and show how Lorentz transformations are seen as conformal transformation in
the celestial sphere. In section 3 we obtain the generic two-point correlator
on the celestial sphere and show in section 4 how the HD correlation function
is recovered. Section 5 extends our computation of the overlap function to the
so-called short-arm detectors. Section 6 contains our conclusions. The paper
contains also three Appendices where some technical details are added.

##  2 The Lorentz group, inertial observers and the celestial sphere

Let us first review the notion of the celestial sphere of an inertial observer
in Minkowski spacetime [12, 13, 14]. The knowledgeable reader can skip this
section.

The celestial sphere is an imaginary sphere of practically infinite radius
centered at the position of an inertial observer. We wish to show now that
Lorentz transformations between different inertial observers correspond to
conformal transformations of the celestial sphere. First, we recall the
stereographic projection and projective coordinates of the two-dimensional
sphere $S^{2}$. The latter is defined as

$\displaystyle S^{2}=\\{(X,Y,Z)\in\mathbb{R}^{3}:X^{2}+Y^{2}+Z^{2}=1\\}.$
(2.1)

The points $(0,0,1)$ and $(0,0,-1)$ are the north $(N)$ and the south ($S$)
poles of the sphere. The $(x,y)$ plane at $z=0$ can be identified with
$\mathbb{C}$ by defining $\zeta=x+iy$. The stereographic projection
$\zeta:S^{2}\backslash\\{N\\}\to\mathbb{C}$ is then the map that associates to
every point $(X,Y,Z)\in S^{2}\backslash\\{N\\}$ the intersection $\zeta$ of
the line which connects the north pole $N$ with $(x,y,z=0$).

Figure 1: The stereographic projection and projective coordinates of the two-
dimensional sphere $S^{2}$.

In other words, we have

$\displaystyle\zeta=x+iy=\frac{X+iY}{1-Z}.$ (2.2)

If we introduce standard spherical coordinates in $\mathbb{R}^{3}$

$\displaystyle X=r\sin\theta\,\cos\phi,\quad Y=r\sin\theta\,\sin\phi,\quad
Z=r\cos\theta,$ (2.3)

we find that Eq. (2.2) is written as

$\displaystyle\zeta(\theta,\phi)=\frac{\sin\theta}{1-\cos\theta}e^{i\phi}.$
(2.4)

Therefore, we get

$\displaystyle{\rm d}\zeta=-\frac{1}{1-\cos\theta}e^{i\phi}{\rm
d}\theta+i\frac{\sin\theta}{1-\cos\theta}e^{i\phi}{\rm d}\phi,$ (2.5)

so that

$\displaystyle{\rm d}\zeta{\rm
d}\overline{\zeta}=\frac{1}{(1-\cos\theta)^{2}}{\rm d}s_{2}^{2},$ (2.6)

where

${\rm d}s_{2}^{2}={\rm d}\theta^{2}+\sin\theta^{2}{\rm d}\phi^{2}$ (2.7)

is the standard metric on the unit $S^{2}$. Since

$\displaystyle\frac{1}{2}(1-\cos\theta)=\frac{1}{1+\zeta\overline{\zeta}},$
(2.8)

the metric on the $S^{2}$ is written as

$\displaystyle{\rm d}s_{2}^{2}=\frac{4}{(1+\zeta\overline{\zeta})^{2}}{\rm
d}\zeta{\rm d}\overline{\zeta}.$ (2.9)

Note in the above construction, the north pole is mapped nowhere. However, by
extending $\mathbb{C}$ to $\mathbb{C}\\!\cup\\!\\{\infty\\}$, the north pole
is mapped to itself and therefore, we can identify the whole of $S^{2}$ with
$\mathbb{C}\\!\cup\\!\\{\infty\\}$. Therefore, complex analytic automorphisms
$f:S^{2}\to S^{2}$ are represented by homomorphic mappings
$g:\mathbb{C}\\!\cup\\!\\{\infty\\}\to\mathbb{C}\\!\cup\\!\\{\infty\\}$. In
particular, it is easy to find that

$\displaystyle{\rm
d}s_{2}^{2}(g(z))=\frac{g^{\prime}\overline{g}^{\prime}(1+\zeta\overline{\zeta})^{2}}{(1+g\overline{g})^{2}}{\rm
d}s_{2}^{2}(\zeta),$ (2.10)

and therefore, complex analytic automorphisms of $S^{2}$ are conformal
transformations. In fact, it can be shown that the conformal orientation-
preserving automorphisms of $S^{2}$ form a group isomorphic to the group of
all complex analytic automorphisms of $S^{2}$. The latter are generated by the
composition of inversion and the linear transformation

$\displaystyle\zeta\to\frac{1}{\zeta},\qquad\zeta\to
a\zeta+b,~{}~{}~{}~{}(a\neq 0)$ (2.11)

and therefore, complex analytic automorphism of $S^{2}$ are the Möbius
transformations

$\displaystyle\zeta\to\frac{a\zeta+b}{c\zeta+d},$ (2.12)

which form a group isomorphic to $SL(2,\mathbb{C})/\mathbb{Z}_{2}$. Thus, the
group of Möbius transformations is the group of all complex analytic
automorphisms of $S^{2}$

$\displaystyle\zeta\to g(\zeta)=\frac{a\zeta+b}{c\zeta+d},\qquad ab-cd=1.$
(2.13)

On the other hand, it is known that the proper orthochronous Lorentz
transformations $O_{+}^{\uparrow}(1,3)$ is isomorphic to the Möbius group
$SL(2,\mathbb{C})/\mathbb{Z}_{2}$ [12, 13]. A convenient set of generators of
the latter are

$\displaystyle L_{i}=\frac{1}{2}\sigma_{i},\qquad
K_{i}=-\frac{i}{2}\sigma_{i},$ (2.14)

where $\sigma_{i}$ ($i=1,2,3)$ are the Pauli matrices. The generators
$L_{i},K_{i}$ satisfy the commutation relations of the $SL(2,\mathbb{C})$
algebra

$\displaystyle[K_{i},K_{j}]=-\epsilon_{ijk}L_{k},\quad[L_{i},L_{j}]=\epsilon_{ijk}L_{k},\quad[K_{i},L_{j}]=\epsilon_{ijk}K_{k}.$
(2.15)

Due to the aforementioned isomorphism, one can prove that boosts in the
$z$-direction in Minkowski spacetime by a hyperbolic angle $\xi$ generated by
$K_{3}$, corresponds to the conformal (Möbius) transformation $\zeta\to
e^{\xi}\zeta$. In analogy, the generators $K_{1},K_{2}$ of $SL(2,\mathbb{C})$
generate boosts in the $x$ and $-y$ directions. Similarly, $L_{1},L_{2}$ and
$L_{3}$, generate rotations around the $-x$, $y$ and $-z$ directions,
respectively. In general, there exists a map
$f:SL(2,\mathbb{C})/\mathbb{Z}_{2}\to O_{+}^{\uparrow}(1,3)$ given explicitly
by

$\displaystyle SL(2,\mathbb{C})\ni S=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$
(2.16)

which implies

$\displaystyle f(S)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{1}{2}\left(|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}\right)&-{\rm
Re}\left(a\overline{b}+c\overline{d}\right)&{\rm
Im}\left(a\overline{b}+c\overline{}\right)&\frac{1}{2}\left(|a|^{2}-|b|^{2}+|c|^{2}-|d|^{2}\right)\\\
-{\rm Re}\left(\overline{a}c+\overline{b}d\right)&{\rm
Re}\left(\overline{a}d+\overline{b}c\right)&-{\rm
Im}\left(a\overline{d}-b\overline{c}\right)&-{\rm
Re}\left(\overline{a}c-\overline{b}d\right)\\\ {\rm
Im}\left(\overline{a}c+\overline{b}d\right)&-{\rm
Im}\left(\overline{a}d+\overline{b}c\right)&{\rm
Re}\left(a\overline{d}-b\overline{c}\right)&{\rm
Im}\left(\overline{a}c-\overline{b}d\right)\\\
\frac{1}{2}\left(|a|^{2}+|b|^{2}-|c|^{2}-|d|^{2}\right)&-{\rm
Re}\left(a\overline{b}-c\overline{d}\right)&{\rm
Im}\left(a\overline{b}-c\overline{}\right)&\frac{1}{2}\left(|a|^{2}-|b|^{2}-|c|^{2}+|d|^{2}\right)\end{pmatrix}.$

One concludes that

Lorentz transformations between inertial observers are conformal motions of
the two-sphere.

###  2.1 Lorentz transformations and the celestial sphere

The sphere described above is actually a spatial sphere, that is a sphere at
radius $r^{2}=x^{2}+y^{2}+z^{2}$. However, we are interested in the sphere
that an observer looks at, that is the celestial sphere. This is not a spatial
sphere defined by $r=$constant, since ingoing, radial light-rays emitted by
the sphere need a time $r$ to arrive from the sphere to the observer at the
origin $r=0$, (we have set the speed of light to unity). Instead, one can
parametrize the emission time of radial ingoing light rays by the value of the
advanced time coordinate $u=t+r$. In other words, we specify events in
spacetime by using the Bondi coordinates $(u,r,\zeta)$. A sphere at constant
$r>0$ and constant $u$ in Bondi coordinates, is the sphere seen by an observer
at $r=0$ and time $u$. The celestial sphere at time $u$ is then the sphere
located at an infinite distance $r\to\infty$, and at a fixed $u$, and it is
the sphere formed by all directions towards which an observer sitting at $r=0$
may look.

In terms of the Bondi coordinates, the metric of the flat Minkowski spacetime
is written as

$\displaystyle{\rm d}s^{2}=-{\rm d}u^{2}+2{\rm d}u{\rm d}r+r^{2}{\rm
d}s_{2}^{2},$ (2.18)

where ${\rm d}s_{2}^{2}$ in the metric on the unit $S^{2}$ given in Eq. (2.9).
Isometries of Minkowski spacetime are just Lorentz transformations, which act
linearly on the flat Cartesian coordinates $x^{\mu}$. On the other hand,
Lorentz tranformations act highly non-linear on the Bondi coordinates.
However, they have a simpler form at large $r$ and in particular, one finds
that under the Lorentz transformations given in Eq. (LABEL:lS), the Bondi
coordinates transform as
$(u,r,\zeta)\to(u^{\prime},r^{\prime},\zeta^{\prime})$ where

$\displaystyle r^{\prime}=F(\zeta,\overline{\zeta})\,r+\mathcal{O}(1),$
(2.19a) $\displaystyle
u^{\prime}=\frac{u}{F(\zeta,\overline{\zeta})}+\mathcal{O}(r^{-1}),$ (2.19b)
$\displaystyle\zeta^{\prime}=\frac{a\zeta+b}{c\zeta+d}+\mathcal{O}(r^{-1}),$
(2.19c)

and

$\displaystyle
F(\zeta,\overline{\zeta})=\frac{|a\zeta+b|^{2}+|c\zeta+d|^{2}}{1+\zeta\overline{\zeta}}.$
(2.20)

Eq.(2.19c) is clearly, to leading order, the standard Möbius transformations
on $S^{2}$ and therefore,

Lorentz transformations are conformal transformations of the celestial sphere.

In particular, by using Eq. (LABEL:lS), we find that the matrix

$\displaystyle SL(2,\mathbb{C})/\mathbb{Z}_{2}\ni
S_{0}=\begin{pmatrix}e^{-\xi/2}&0\\\ 0&e^{\xi/2}\end{pmatrix},$ (2.21)

corresponds to the $O_{+}^{\uparrow}(1,3)$ matrix

$\displaystyle L_{b}=\begin{pmatrix}\cosh\xi&0&0&-\sinh\xi\\\ 0&1&0&0\\\
0&0&1&0\\\ -\sinh\xi&0&0&\cosh\xi\end{pmatrix},$ (2.22)

which is nothing else than a Lorentz boost with rapidity

$\displaystyle e^{\xi}=\gamma(1-\vec{v}\cdot\widehat{n}),$ (2.23)

along the $z$-direction. In Eq. (2.23), the unit vector
$\widehat{n}=(n_{1},n_{2},n_{3})$ ($n_{i}n^{i}=1$) is $\widehat{e}_{z}$. Then,
$F(\zeta,\overline{\zeta})$ turns out to be

$\displaystyle
F(\zeta,\overline{\zeta})=\gamma(1-\vec{v}\cdot\widehat{n}),\qquad\gamma=(1-v^{2})^{-1/2}$
(2.24)

and therefore, Lorentz boosts corresponds to the following transformations of
the Bondi coordinates

$\displaystyle u^{\prime}$ $\displaystyle=\Omega(\widehat{n})u,$
$\displaystyle r^{\prime}$ $\displaystyle=\frac{r}{\Omega(\widehat{n})},$
$\displaystyle\zeta^{\prime}$ $\displaystyle=\Omega(\widehat{n})\zeta,$ (2.25)

where

$\displaystyle\Omega(\widehat{n})=\frac{1}{\gamma(1-\vec{v}\cdot\widehat{n})},$
(2.26)

is the Doppler shift factor. It is easy to verify that Eq.(2.25) gives rise to
the aberration formula for outgoing light rays

$\displaystyle\cot\frac{\theta^{\prime}}{2}=\sqrt{\frac{1-v}{1+v}}\cot\frac{\theta}{2}.$
(2.27)

Then by using Eq. (2.27), we find that under a Lorentz boost in the
$\widehat{n}$ direction, the metric of the celestial sphere transforms as

$\displaystyle{\rm d}\theta^{{}^{\prime}2}+\sin^{2}\theta^{\prime}{\rm
d}\phi^{{}^{\prime}2}=\Omega^{2}(\widehat{n})\Big{(}{\rm
d}\theta^{2}+\sin^{2}\theta{\rm d}\phi^{2}\Big{)}.$ (2.28)

##  3 The two- and three-point Correlators on the Celestial Sphere

The goal of this section is two explain how a generic two-point correlator can
be written on the celestial sphere.

The projective coordinate $\zeta$ in $\mathbb{C}\\!\cup\\!\\{\infty\\}$ can be
parametrized in terms of the unit vector $\widehat{n}$ as

$\displaystyle\zeta=\frac{n^{1}+in^{2}}{1-n^{3}},$ (3.1)

so that

$\displaystyle
n^{1}=\frac{\zeta+\overline{\zeta}}{1+\zeta\overline{\zeta}},\qquad
n^{2}=\frac{1}{i}\frac{\zeta-\overline{\zeta}}{1+\zeta\overline{\zeta}},\qquad
n^{3}=-\frac{1-\zeta\overline{\zeta}}{1+\zeta\overline{\zeta}},$ (3.2)

and the metric on the $S^{2}$ becomes

$\displaystyle{\rm d}s_{2}^{2}={\rm d}\widehat{n}\cdot{\rm d}\widehat{n}={\rm
d}n^{i}{\rm d}n^{i},\qquad i=1,2,3.$ (3.3)

Notice that since the $S^{2}$ is formed by light rays which satisfy

$\displaystyle{x^{0}}x^{0}-\vec{x}\cdot\vec{x}=0,$ (3.4)

the unit vector can be expressed as

$\displaystyle n^{i}=\frac{x^{i}}{x^{0}},\qquad x^{0}=|\vec{x}|>0.$ (3.5)

Clearly, $\widehat{n}$ transforms as a vector under space rotations, whereas,
under Lorentz boosts, which act linearly on the Minkowski coordinates
$x^{\mu}$, it is easy to find, using the transformation properties of
$(x^{0},\vec{x})$

$\displaystyle x^{0}{}^{\prime}$
$\displaystyle=\gamma\Big{(}x^{0}-\vec{v}\cdot\vec{x}\Big{)},$
$\displaystyle\vec{x}\,{}^{\prime}$
$\displaystyle=\vec{x}+(\gamma-1)\widehat{v}\,\vec{x}\cdot\widehat{v}-\gamma
x^{0}\vec{v},$ (3.6)

that $n^{i}$ transforms as

$\displaystyle\widehat{n}\to\widehat{n}^{\prime}=\Omega(\widehat{n})\Big{(}\widehat{n}+(\gamma-1)\widehat{v}(\widehat{v}\cdot\widehat{n})-\gamma\vec{v}\Big{)}.$
(3.7)

For an infinitesimal Lorentz boost generated by $K$, using Eq. (2.26) we get

$\displaystyle\delta_{K}\widehat{n}=-\vec{v}+(\widehat{v}\cdot\widehat{n})\widehat{n},\qquad\delta_{K}\Omega=\vec{v}\cdot\widehat{n}.$
(3.8)

It is not difficult to verify that indeed, Lorentz boosts that transform the
unit vector $\widehat{n}$ as in Eq. (3.7), act like conformal transformations
on $S^{2}$ since

$\displaystyle{\rm d}\widehat{n}\cdot{\rm d}\widehat{n}\to{\rm
d}\widehat{n}^{\prime}\cdot{\rm
d}\widehat{n}^{\prime}=\Omega^{2}(\widehat{n}){\rm d}\widehat{n}\cdot{\rm
d}\widehat{n}.$ (3.9)

Let us now consider the two-point correlator
$C_{2}(\vec{x}_{1},\vec{x}_{2})\Big{|}_{S^{2}}=C_{2}(\widehat{n}_{1},\widehat{n}_{2})$
at equal time $t$ and for points $\vec{x}_{1}$ and $\vec{x}_{2}$ both on the
$S^{2}$. We will demand invariance under the Lorentz group. Rotational
invariance leads to dependence on the distance

$\displaystyle d_{12}^{2}=|\vec{x}_{1}-\vec{x}_{2}|^{2}=4\sigma,$ (3.10)

where

$\displaystyle\sigma=\frac{1}{2}(1-\widehat{n}_{1}\cdot\widehat{n}_{2})$
(3.11)

is the geodesic invariant distance between points on $S^{2}$. Therefore, we
should have

$C_{2}=C_{2}(\sigma,t;w),$ (3.12)

where $w$ is the scaling dimension of $C_{2}$ since Lorentz boosts are
actually conformal rescalings of the $S^{2}$. In particular, we find that
under Lorentz boosts, $\sigma$ transforms as

$\displaystyle\sigma\to\sigma^{\prime}=\Omega(\widehat{n}_{1})\Omega(\widehat{n}_{2})\sigma,$
(3.13)

or, in infinitesimal form,

$\displaystyle\delta_{K}\sigma=(\vec{v}\cdot\widehat{n}_{1}+\vec{v}\cdot\widehat{n}_{2})\sigma.$
(3.14)

In addition, if the scaling dimension of $C_{2}$ is $w$, we will have that

$\displaystyle C_{2}(\sigma^{\prime};w)\to
C_{2}^{\prime}(\sigma,t;w)=\Omega(\widehat{n}_{1})^{-w}\Omega(\widehat{n}_{2})^{-w}C_{2}(\sigma;w).$
(3.15)

Therefore, the change in the two-point correlator $C_{2}$ under Lorentz boosts
is

$\displaystyle\delta_{K}C_{2}(\sigma,t;w)=w\Big{(}\delta_{K}\Omega(\widehat{n}_{1})+\delta_{K}\Omega(\widehat{n}_{2})\Big{)}C_{2}(\sigma,t;w)+\delta_{K}\sigma\frac{{\rm
d}C_{2}(\sigma,t;w)}{{\rm d}\sigma}.$ (3.16)

Hence, we find that

$\displaystyle\delta_{K}C_{2}=\left(wC_{2}+\sigma\frac{{\rm d}C_{2}}{{\rm
d}\sigma}\right)(\vec{v}\cdot\widehat{n}_{1}+\vec{v}\cdot\widehat{n}_{2}).$
(3.17)

Invariance under Lorentz boosts ($\delta_{K}C_{2}=0$) then leads to

$\displaystyle wC_{2}+\sigma\frac{{\rm d}C_{2}}{{\rm d}\sigma}=0,$ (3.18)

and therefore,

$\displaystyle C_{2}(\sigma;w)=\frac{C_{02}(w)}{\sigma^{w}}.$ (3.19)

In particular, for $w=0$ we get (as a limit of Eq. (3.19))

$\displaystyle C_{2}(\sigma,0)=\alpha_{2}\ln\sigma+\beta_{2},$ (3.20)

where $\alpha_{2}$ and $\beta_{2}$ are constants.

##  4 Hellings and Downs from Lorentz invariance or conformal symmetry on the
celestial sphere

We now come to the HD correlation function. The goal of this section is to
demonstrate that it can be understood in terms of Lorentz transformations or
conformal transformations on the celestial sphere. Let us consider metric
perturbations which, for the most general GW is a superposition of
(transverse-traceless) plane waves, i.e.,

$\displaystyle
h_{ij}^{TT}(t,\vec{x})=\sum_{A=+,\times}\int_{-\infty}^{\infty}{\rm
d}f\int{\rm
d}^{2}\widehat{n}\,h_{A}(f,\widehat{n})e^{A}_{ij}(\widehat{n})e^{-2\pi
if(t-\widehat{n}\cdot\vec{x})},$ (4.1)

where $A=+,\times$ are the polarizations of the GWs, $\widehat{n}$ their
direction of propagation and $f$ their frequency. The polarization tensors
$e^{A}_{ij}(\widehat{n})$ are given by

$\displaystyle
e^{+}_{ij}=\widehat{u}_{i}\widehat{u}_{j}-\widehat{w}_{i}\widehat{w}_{j},\qquad
e^{\times}_{ij}=\widehat{u}_{i}\widehat{w}_{j}+\widehat{w}_{i}\widehat{u}_{j},$
(4.2)

where the vectors $\widehat{n},\widehat{u}$ and $\widehat{w}$ are explicitly
given by

$\displaystyle\widehat{n}$
$\displaystyle=(-\sin\theta\cos\phi,-\sin\theta\cos\phi,-\cos\theta),$
$\displaystyle\widehat{u}$ $\displaystyle=(\sin\phi,-\cos\phi,0),$
$\displaystyle\widehat{w}$
$\displaystyle=(-\cos\theta\cos\phi,-\cos\theta\sin\phi,\sin\theta),$ (4.3)

and depicted in Fig. 2.

Figure 2: The GW source is in the $-\widehat{n}$ direction and the GW
propagates towards the origin.

In addition, we will need later the ensemble average of a stationary,
isotropic and unpolarized, stochastic GW background, which is characterized by

$\displaystyle\langle
h^{*}_{A_{1}}(f_{1},\widehat{n}_{1})h_{A_{2}}(f_{2},\widehat{n}_{2})\rangle=\delta(f_{1}-f_{2})\frac{\delta^{(2)}(\widehat{n}_{1}-\widehat{n}_{2})}{4\pi}\delta_{A_{1}A_{2}}\frac{1}{2}S(f_{1}),$
(4.4)

where $S(f)$ is the spectral density of the stochastic background.

Let us now recall how the periodicity of the pulses of a pulsar is affected by
the passing GW. If the pulsar is at position
$\vec{x}_{a}=d_{a}\widehat{n}_{a}$, then the change in the periodicity (time
residual)

$z_{a}=\left(\frac{\Delta T}{T}\right)_{a}$

due to a GW $h_{ij}^{TT}(t,\vec{x})$ as seen by an observer at $\vec{x}=0$ is

$\displaystyle
z_{a}=\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}\bigg{(}h_{ij}^{TT}(t,\widehat{n},\vec{x}=0)-h_{ij}^{TT}(t-d_{a}/c,\widehat{n}_{a},\vec{x}_{a})\bigg{)}.$
(4.5)

The term in the parenthesis represents the difference between the metric
perturbation which arrive at the position of the observer at $\vec{x}=0$ and
the position of the pulsar at $\vec{x}_{a}$. As it is standard in any similar
consideration, we will ignore the pulsar metric perturbation [15] and write
instead

$\displaystyle
z_{a}(t,\widehat{n})=\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}h_{ij}^{TT}(t,\widehat{n},\vec{x}=0).$
(4.6)

A stochastic GW background can be constructed by the superposition of plane GW
of all possible frequencies from all possible directions. Therefore, the
change in the periodicity of the pulsar due to a stochastic GW background can
be expressed as

$\displaystyle
z_{a}(t)=\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}\sum_{A}\int_{-\infty}^{+\infty}{\rm
d}f\int\frac{{\rm
d}^{2}{\widehat{n}}}{1-\widehat{n}\cdot\widehat{n}_{a}}h_{A}(f,\widehat{n})e_{ij}^{A}(\widehat{n})\,e^{-2\pi
ift},$ (4.7)

where Eq. (4.1) has been used. Then by using Eq. (4.4), we find that the
correlator of time residuals $\langle z_{a}z_{b}\rangle$ of two pulsars at
directions $\widehat{n}_{a}$ and $\widehat{n}_{b}$ is given by

$\displaystyle\langle z_{a}z_{b}\rangle=\frac{1}{2}\int_{-\infty}^{\infty}{\rm
d}f\,S(f)\,\Gamma_{ab}(\widehat{n}_{a},\widehat{n}_{b})$ (4.8)

where we have defined the overlap function

$\displaystyle\Gamma_{ab}(\widehat{n}_{a},\widehat{n}_{b})=\int\frac{{\rm
d}^{2}\widehat{n}}{4\pi}\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}\frac{\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}}{1-\widehat{n}\cdot\widehat{n}_{b}}\sum_{A=+,\times}e^{A}_{ij}(\widehat{n})e^{A}_{kl}(\widehat{n}).$
(4.9)

We are interested in the transformation properties of the correlator
$\Gamma_{ab}$ in Eq.(4.9) under Lorentz transformations. We have seen that
three-dimensional rotational invariance requires that the correlator is a
function of $\sigma_{ab}$ only

$\Gamma_{ab}=\Gamma_{ab}(\sigma_{ab}),$ (4.10)

that is a function of the angular separation of the pulsars. It remains to
determine the transformation properties of $\Gamma_{ab}$ under Lorentz boosts.
We will consider a Lorentz boost of velocity $\vec{v}$ in the direction of the
GW, i.e., $\vec{v}=\widehat{n}$. Notice that this particular choice does not
spoil the generality of the following analysis, since we need to integrate
over all directions $\widehat{n}$ to account for a stochastic GW background.
This is equivalent to keeping $\widehat{n}$ fixed and boosting in all possible
directions. To determine the transformation law of $\Gamma_{ab}$, we recall
Eqs.(3.7), (3.11) and (3.13) so that

$\displaystyle\widehat{n}_{a}^{\prime
i}=\Omega(\widehat{n}_{a})\Big{(}\widehat{n}_{a}^{i}+\widehat{v}^{i}(\widehat{v}\cdot\widehat{n}_{a})(\gamma-1)-\gamma
v^{i}\Big{)},$ (4.11)

and

$\displaystyle
1-\widehat{n}^{\prime}\cdot\widehat{n}_{a}^{\prime}=\Omega(\widehat{n})\Omega(\widehat{n}_{a})(1-\widehat{n}\cdot\widehat{n}_{a}).$
(4.12)

The polarization tensors $e^{A}_{ij}$ are invariant under Lorentz boosts along
the direction of propagation. Indeed, in this case,
$(\widehat{u},\widehat{w})$ are invariant and so are the polarization tensors.
In the Newman-Penrose formalism, $(\widehat{u},\widehat{w})$ span the
$(m^{\alpha},\overline{m}^{a}$) plane, which is invariant under Lorentz boosts
in the $(\ell^{a},n^{a})$ directions. Thus, using Eq.(4.11) we find that

$\displaystyle\widehat{n}_{a}^{{}^{\prime}i}\widehat{n}_{a}^{{}^{\prime}j}e^{A^{\prime}}_{ij}(\widehat{n}^{\prime})=\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}e^{A}_{ij}(\widehat{n}).$
(4.13)

Under a Lorentz boost, the volume element on the $S^{2}$ in Eq. (4.9)
transforms as

$\displaystyle{\rm d}^{2}\widehat{n}\to{\rm
d}^{2}\widehat{n}^{\prime}\Omega^{2}(\widehat{n}).$ (4.14)

The amplitude of the GW is also invariant as follows from the transformation
of the Weyl scalar $\Psi_{4}$ under Lorentz boosts [16]

$\displaystyle\Psi_{4}^{\prime}=\Omega^{-2}(\widehat{n})\Psi_{4}.$ (4.15)

Since [17]

$\displaystyle\Psi_{4}=\frac{\omega^{2}}{2}\Big{(}h_{+}^{2}-ih_{\times}^{2}\Big{)},$
(4.16)

and $\omega=2\pi f$ transforms as

$\displaystyle\omega^{\prime}=\Omega^{-1}(\widehat{n})\omega,$ (4.17)

we get that the amplitudes $h_{A}$ are invariant under Lorentz boosts.
Therefore, for a stationary, isotropic and unpolarized stochastic GW
background, the ensemble average in Eq. (4.4) is invariant

$\displaystyle\langle
h^{*}_{A_{1}}(f_{1},\widehat{n}_{1})h_{A_{2}}(f_{2},\widehat{n}_{2})\rangle=\langle
h^{*^{\prime}}_{A_{1}}(f_{a}^{\prime},\widehat{n}_{1}^{\prime})h^{\prime}_{A_{2}}(f_{2}^{\prime},\widehat{n}_{2}^{\prime})\rangle$
(4.18)

or, in other words,

$\displaystyle\delta(f_{1}^{\prime}-f_{2}^{\prime})\frac{\delta^{(2)}(\widehat{n}_{1}^{\prime}-\widehat{n}_{2}^{\prime})}{4\pi}\delta_{A_{1}A_{2}}\frac{1}{2}S^{\prime}(f_{1}^{\prime})=\delta(f_{1}-f_{2})\frac{\delta^{(2)}(\widehat{n}_{1}-\widehat{n}_{2})}{4\pi}\delta_{A_{1}A_{2}}\frac{1}{2}S(f_{1})$
(4.19)

Since $\delta^{(2)}(\widehat{n}_{1}-\widehat{n}_{2})$ and
$\delta(f_{1}-f_{2})$ transform as

$\displaystyle\delta^{(2)}(\widehat{n}_{1}-\widehat{n}_{2})\to\Omega(-\widehat{n}_{1})^{2}\delta^{(2)}(\widehat{n}_{1}-\widehat{n}_{2}),\qquad\delta(f_{1}-f_{2})\to\Omega(\widehat{n}_{1})\delta(f_{1}-f_{2}),$
(4.20)

respectively, we get that the spectral density of the stochastic background
transforms as

$\displaystyle S^{\prime}(f_{1})\to\Omega(\widehat{n}_{1})S(f_{1}).$ (4.21)

and therefore, since ${\rm d}f$ transforms as

${\rm d}f\to{\rm d}f^{\prime}=\Omega(\widehat{n}){\rm d}f,$ (4.22)

we find that ${\rm d}fS(f)$ is invariant

$\displaystyle{\rm
d}f^{\prime}S^{\prime}(f^{\prime})=\Omega(\widehat{n})^{-1}{\rm
d}f\,\Omega(\widehat{n})S(f)={\rm d}fS(f).$ (4.23)

This can also be deduced from the relation

$\displaystyle\langle h_{ij}h^{ij}\rangle=4\int_{0}^{\infty}{\rm d}f\,S(f).$
(4.24)

Since the left-hand side of Eq. (4.24) is invariant, so is the right-hand
side.

Putting all together, it is easy to verify that overlap function transforms
under Lorentz boosts as

$\displaystyle\Gamma_{ab}\to\Gamma^{\prime}_{ab}$
$\displaystyle=\int\frac{{\rm
d}^{2}\widehat{n}}{4\pi}\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}\frac{\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}}{1-\widehat{n}\cdot\widehat{n}_{b}}\Omega(\widehat{n}_{a})\Omega(\widehat{n}_{b})\sum_{A=+,\times}e^{A}_{ij}(\widehat{n})e^{A}_{kl}(\widehat{n}).$
(4.25)

However, this transformation has fixed points since there are Lorentz boosts
that leave $\Gamma_{ab}$ invariant. Indeed, since

$\displaystyle\Omega(\widehat{v})\Omega(-\widehat{v})=1,$ (4.26)

for pulsars located at opposite directions, $\widehat{n}_{a}$ and
$\widehat{n}_{b}=-\widehat{n}_{a}$, a Lorentz boost along their direction (so
that $\widehat{v}=\pm\widehat{n}_{a}$), leaves their correlator
$\Gamma_{v}=\Gamma_{ab}(\pm v,\mp v)$ invariant

$\displaystyle\Gamma_{v}^{\prime}=\Gamma_{v}.$ (4.27)

Then, the reduced overlap function $\Gamma_{ab}^{-}=\Gamma_{ab}-\Gamma_{v}$
transforms as in Eq. (4.25) without fixed points and therefore,

$\displaystyle\overline{\Gamma}_{ab}=\frac{\Gamma^{-}_{ab}}{\sigma_{ab}}=\int\frac{{\rm
d}^{2}\widehat{n}}{4\pi}\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}\frac{\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}}{1-\widehat{n}\cdot\widehat{n}_{b}}\frac{1}{\sigma_{ab}}\sum_{A=+,\times}e^{A}_{ij}(\widehat{n})e^{A}_{kl}(\widehat{n})$
(4.28)

is invariant under Lorentz boosts

$\displaystyle\overline{\Gamma}^{\prime}_{ab}=\overline{\Gamma}_{ab}.$ (4.29)

Thus, allowing logarithmic scaling violation (see Appendix A for a technical
explanation), by using Eq. (3.20), we find

$\displaystyle\overline{\Gamma}_{ab}=\alpha\,\ln\sigma_{ab}+\beta.$ (4.30)

from where we get the functional form of the HD curve

$\Gamma_{ab}=\alpha\,\sigma_{ab}\ln\sigma_{ab}+\beta\,\sigma_{ab}+\Gamma_{v}$.

(4.31)

###  4.1 Recovering the Hellings and Downs correlation

Our goal now is to determine the constants $\alpha$, $\beta$ and $\Gamma_{v}$
of Eq. (4) and to recover the HD correlation function. For this we need two
conditions:

  1. 1.

PTA detectors operate in the “long-arm” (or short-wavelength) limit in which
the distance between the pulsars and the earth is much longer than the
wavelengths of the GWs; this induces a well-defined relation of the
correlators for pulsars in the same and in opposite directions;

  2. 2.

the vanishing of the integral of the two-point correlator over the whole
celestial sphere basically due to the isotropy of the stochastic GW background
and its traceless and transverse properties.

First, as shown in Appendix B, by using the fact that the two-point correlator
of two pulsars separated by $\pi$ ($\sigma_{ab}=1$) is half that for $0$
angular separation ($\sigma=0$), we get that

$\displaystyle\Gamma_{ab}=\alpha\,\sigma_{ab}\ln\sigma_{ab}+\beta\,\sigma_{ab}-2\beta.$
(4.32)

Secondly, as shown in Appendix C, one can prove that the integral of the two-
point function over the whole celestial sphere should vanish and therefore

$\displaystyle\int{\rm d}^{2}\Omega_{2}\,\Gamma_{ab}=-4\pi\int_{0}^{1}{\rm
d}\sigma_{ab}\Big{(}\alpha\,\sigma_{ab}\ln\sigma_{ab}+\beta\,\sigma_{ab}-2\beta\Big{)}=\pi(\alpha+6\beta)=0,$
(4.33)

This leads to

$\displaystyle\beta=-\frac{\alpha}{6}.$ (4.34)

Knowing that the overall normalization, that is the coefficient $\alpha$ is
arbitrary [11], setting $\alpha=1$, standard form of the HD curve is recovered

$\Gamma_{ab}=\sigma_{ab}\ln\sigma_{ab}-\dfrac{1}{6}\,\sigma_{ab}+\dfrac{1}{3}$,

(4.35)

is obtained. Choosing $\alpha=3/2$, one would have got the alternative, but
equivalent, HD function used in the literature

$\Gamma_{ab}=\dfrac{3}{2}\sigma_{ab}\ln\sigma_{ab}-\dfrac{1}{4}\,\sigma_{ab}+\dfrac{1}{2}$.

(4.36)

###  4.2 Going beyond the Hellings and Downs correlation: the three-point
correlator

Conformal symmetry on the celestial sphere can also specify (up to a
multiplicative constant) the three-point correlator. Indeed, applying similar
considerations as in the previous section, the three-point correlator
$C_{3}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})\Big{|}_{S^{2}}=C_{3}(\widehat{n}_{1},\widehat{n}_{2},\widehat{n}_{3})$
is again, due to rotational symmetry, a function of the geodesic invariant
distances $\sigma_{ij}=|\vec{x}_{i}-\vec{x}_{j}|^{2}/4$, ($i,j=1,2,3$) so that
$C_{3}=C_{3}(\sigma_{12},\sigma_{23},\sigma_{13})$. Let us further assume that
under Lorentz boosts (conformal transformations), the three-point correlator
$C_{3}$ of fields of dimension $w$ transforms as

$\displaystyle\delta_{K}C_{3}$
$\displaystyle=w\Big{(}\delta_{K}\Omega(\widehat{n}_{1})+\delta_{K}\Omega(\widehat{n}_{2})+\delta_{K}\Omega(\widehat{n}_{3})\Big{)}C_{3}+\delta\sigma_{12}\frac{\partial
C_{3}}{\partial\sigma_{12}}+\delta\sigma_{23}\frac{\partial
C_{3}}{\partial\sigma_{23}}+\delta\sigma_{13}\frac{\partial
C_{3}}{\partial\sigma_{13}}$
$\displaystyle=w\vec{v}\cdot(\widehat{n}_{1}+\widehat{n}_{2}+\widehat{n}_{3})C_{3}+\vec{v}\cdot(\widehat{n}_{1}+\widehat{n}_{2})\sigma_{12}\frac{\partial
C_{3}}{\partial\sigma_{12}}+\vec{v}\cdot(\widehat{n}_{2}+\widehat{n}_{3})\sigma_{23}\frac{\partial
C_{3}}{\partial\sigma_{23}}$
$\displaystyle+\vec{v}\cdot(\widehat{n}_{1}+\widehat{n}_{3})\sigma_{13}\frac{\partial
C_{3}}{\partial\sigma_{13}}.$ (4.37)

Then, since invariance under Lorentz boosts $(\delta_{K}C_{3}=0)$ should hold
for any $\widehat{n}_{i}$ and any $\vec{v}$, one is led to the equations

$wC_{3}=2\sigma_{12}\frac{\partial
C_{3}}{\partial\sigma_{12}}=2\sigma_{23}\frac{\partial
C_{3}}{\partial\sigma_{23}}=2\sigma_{13}\frac{\partial
C_{3}}{\partial\sigma_{13}},$ (4.38)

which are solved by

$C_{3}(\sigma_{12},\sigma_{23},\sigma_{13},w)=\frac{C_{03}(w)}{\Big{(}\sigma_{12}\sigma_{23}\sigma_{13}\Big{)}^{\frac{w}{2}}}.$
(4.39)

Again, for $w=0$, the three-point correlators
$C_{3}(\sigma_{12},\sigma_{23},\sigma_{13},w)=C_{3}(\sigma,w)$, turns out to
be

$C_{3}(\sigma,0)=\alpha_{3}\,\ln\Big{(}\sigma_{12}\sigma_{23}\sigma_{13}\Big{)}+\beta_{3},$
(4.40)

where $\alpha_{3}$ and $\beta_{3}$ are constants. To proceed, let us assume
that there exists some sort of non-Gaussianity in the stochastic gravitational
wave background. In this case, the three-point correlator of time residuals
$C_{abc}=\langle z_{a}z_{b}z_{c}\rangle$ will be given by

$\displaystyle C_{abc}=$ $\displaystyle\int{\rm d}f_{1}{\rm d}f_{2}{\rm
d}f_{3}\int\frac{{\rm d}^{2}\widehat{n}_{1}}{4\pi}\frac{{\rm
d}^{2}\widehat{n}_{2}}{4\pi}\frac{{\rm
d}^{2}\widehat{n}_{3}}{4\pi}\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}\widehat{n}_{c}^{m}\widehat{n}_{c}^{n}}{(1-\widehat{n}_{1}\cdot\widehat{n}_{a})(1-\widehat{n}_{2}\cdot\widehat{n}_{b})(1-\widehat{n}_{3}\cdot\widehat{n}_{c})}e^{-2\pi
i(f_{1}+f_{2}+f_{3})t}$ $\displaystyle\hskip 28.45274pt\langle
h^{A}(f_{1},\widehat{n}_{1})h^{A^{\prime}}(f_{2},\widehat{n}_{2})h^{A^{\prime\prime}}(f_{3},\widehat{n}_{3})\rangle\sum_{A,A^{\prime},A^{\prime\prime}=+,\times}\epsilon^{A}_{ij}(\widehat{n}_{1})\epsilon^{A^{\prime}}_{kl}(\widehat{n}_{2})\epsilon^{A^{\prime\prime}}_{mn}(\widehat{n}_{3}).$
(4.41)

We are interested in the transformation properties of the above correlator
under Lorentz boosts, that is conformal transformations of the celestial
sphere. By rotational symmetry, the correlator in Eq. (4.41) will depend on
the invariants $\sigma_{ab},~{}\sigma_{bc}$ and $\sigma_{ac}$. By using Eqs.
(4.12), (4.17) and (4.22), it can easily be verified that the three-point
correlator transforms under Lorentz boosts as

$\displaystyle\delta_{K}C_{abc}$
$\displaystyle=-\vec{v}\cdot(\widehat{n}_{a}+\widehat{n}_{b}+\widehat{n}_{c})C_{abc}+\vec{v}\cdot(\widehat{n}_{a}+\widehat{n}_{b})\sigma_{ab}\frac{\partial
C_{abc}}{\partial\sigma_{ab}}+\vec{v}\cdot(\widehat{n}_{b}+\widehat{n}_{c})\sigma_{bc}\frac{\partial
C_{abc}}{\partial\sigma_{bc}}$
$\displaystyle+\vec{v}\cdot(\widehat{n}_{a}+\widehat{n}_{c})\sigma_{ac}\frac{\partial
C_{abc}}{\partial\sigma_{ac}}+\Omega_{abc},$ (4.42)

where

$\displaystyle\Omega_{abc}=$ $\displaystyle-\int{\rm d}f_{1}{\rm d}f_{2}{\rm
d}f_{3}\int\frac{{\rm d}^{2}\widehat{n}_{1}}{4\pi}\frac{{\rm
d}^{2}\widehat{n}_{2}}{4\pi}\frac{{\rm
d}^{2}\widehat{n}_{3}}{4\pi}\dfrac{2(v^{i}\widehat{n}_{a}^{j}\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}\widehat{n}_{c}^{m}\widehat{n}_{c}^{n}+\mbox{cyclic})}{(1-\widehat{n}_{1}\cdot\widehat{n}_{a})(1-\widehat{n}_{2}\cdot\widehat{n}_{b})(1-\widehat{n}_{3}\cdot\widehat{n}_{c})}e^{-2\pi
i(f_{1}+f_{2}+f_{3})t}$ (4.43) $\displaystyle\hskip 28.45274pt\langle
h^{A}(f_{1},\widehat{n}_{1})h^{A^{\prime}}(f_{2},\widehat{n}_{2})h^{A^{\prime\prime}}(f_{3},\widehat{n}_{3})\rangle\sum_{A,A^{\prime},A^{\prime\prime}=+,\times}\epsilon^{A}_{ij}(\widehat{n}_{1})\epsilon^{A^{\prime}}_{kl}(\widehat{n}_{2})\epsilon^{A^{\prime\prime}}_{mn}(\widehat{n}_{3})$
$\displaystyle=-\int{\rm d}f_{1}{\rm d}f_{2}{\rm d}f_{3}\int\frac{{\rm
d}^{2}\widehat{n}_{1}}{4\pi}\frac{{\rm d}^{2}\widehat{n}_{2}}{4\pi}\frac{{\rm
d}^{2}\widehat{n}_{3}}{4\pi}\dfrac{6v^{i}\widehat{n}_{a}^{j}\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}\widehat{n}_{c}^{m}\widehat{n}_{c}^{n}}{(1-\widehat{n}_{1}\cdot\widehat{n}_{a})(1-\widehat{n}_{2}\cdot\widehat{n}_{b})(1-\widehat{n}_{3}\cdot\widehat{n}_{c})}e^{-2\pi
i(f_{1}+f_{2}+f_{3})t}$ $\displaystyle\hskip 28.45274pt\langle
h^{A}(f_{1},\widehat{n}_{1})h^{A^{\prime}}(f_{2},\widehat{n}_{2})h^{A^{\prime\prime}}(f_{3},\widehat{n}_{3})\rangle\sum_{A,A^{\prime},A^{\prime\prime}=+,\times}\epsilon^{A}_{ij}(\widehat{n}_{1})\epsilon^{A^{\prime}}_{kl}(\widehat{n}_{2})\epsilon^{A^{\prime\prime}}_{mn}(\widehat{n}_{3}).$

In the last equality in Eq. (4.43), the invariance of $C_{abc}$ under
$n_{i}\to n_{j}$ ($i,j=1,2,3$) has been used. As in the case of the two-point
correlator, by defining

$\displaystyle\overline{C}_{abc}=\frac{C_{abc}}{(\sigma_{ab}\sigma_{bc}\sigma_{ac})^{1/2}},$
(4.44)

and taking $\widehat{v}=\widehat{n}_{1}$, we find that $\Omega_{abc}=0$. In
addition, it can be easily verified that $\overline{C}_{abc}$ transforms like
a three-point correlator for fields of dimension $w=0$, and therefore,
according to Eq. (4.40), it will be given by

$\displaystyle\overline{C}_{abc}=\alpha_{3}\ln(\sigma_{ab}\sigma_{bc}\sigma_{ac})+\beta_{3}.$
(4.45)

This specifies the three-point correlator of the time residuals to be

$\displaystyle
C_{abc}=(\sigma_{ab}\sigma_{bc}\sigma_{ac})^{1/2}\,\bigg{(}\alpha_{3}\ln(\sigma_{ab}\sigma_{bc}\sigma_{ac})+\beta_{3}\bigg{)},$
(4.46)

It should be noted that both the two- and three-point correlators have the
form of the corresponding correlators of a logarithmic conformal field theory
as discussed in Appendix A.

The Eq. (4.46) is our generic result, where the coefficients depend on the
non-Gaussian nature of the tensor modes. However, even if the GW source signal
is intrinsically non-Gaussian and with a non-vanishing three-point correlator,
one has to account for the fact that the stochastic signal measured by the PTA
collaborations is a sum of the superposition of GWs from a large number of
independent sources. This suppresses the non-Gaussianity of the time
residuals, since it destroys the phase coherence needed to have a nonvanishing
non-Gaussianity. Indeed, consider that the GW propagates in a perturbed
universe and travel for cosmological distances. Each GW will pick up a Shapiro
time delay from the time of emission $t_{e}$

$\delta t=2\int_{t_{e}}^{t}{\rm
d}t^{\prime}\Psi(\vec{x}+\widehat{n}(t^{\prime}-t),t^{\prime}),$ (4.47)

where $\Psi$ is the gravitational potential along the trajectory and $t_{e}$
the emission time. When interfering with the light from the pulsars, the GW
has therefore acquired a phase shift of the order of $f\delta t$ compared to
the propagation in a homogeneous universe. This would not a problem were the
phase shift the same for all the GW measurements. However, if they vary, the
average correlation of waves at three points would pick up a factor of

$\Big{<}{\rm exp}(i\sum_{i}f_{i}\delta t_{i})\Big{>}\sim{\rm
exp}\left(-\sum_{i}f^{2}_{i}\langle\delta t^{2}\rangle\right))\sim{\rm
exp}\left(-10^{-8}\sum_{i}f^{2}_{i}t^{2}\right),$ (4.48)

which gives an exponential suppression of the three-point correlator of the
time residuals when evaluated at the age of the universe [18, 19], unless one
considers flattened triangle configurations in frequency space [20].
Eventually, one might also hope to escape the suppression if the GWs are
produced within the same structure, e.g. by clustered binaries of supermassive
black holes.

##  5 Recovering the overlap function for an array of “short-arm” LIGO-like
detectors

The Hellings and Downs correlation is obtained in the so-called long-arm
approximation, when the wavelengths of the GWs are much smaller than the
distance between the earth and the pulsars. In the opposite limit, the so-
called short-arm, in which the wavelengths are much larger than the arms of
the detectors, e.g., LIGO and Virgo, the overlap function is also easily
recovered by our symmetry arguments as follows.

The time residual is written in the short-arm approximation as

$\displaystyle z_{a}(t)=\int_{-\infty}^{+\infty}{\rm d}f\,f\int{\rm
d}^{2}{\widehat{n}}\,\,\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}\sum_{A}h_{A}(f,\widehat{n})\,e_{ij}^{A}(\widehat{n})\,e^{-2\pi
ift},$ (5.1)

and therefore, the correlator $\langle z_{a}z_{b}\rangle$ turns out to be in
this case

$\displaystyle\langle z_{a}z_{b}\rangle=\frac{1}{2}\int_{-\infty}^{\infty}{\rm
d}f\,S(f)\,\Gamma_{ab}^{f}(\widehat{n}_{a},\widehat{n}_{b}),$ (5.2)

where the overlap function is given by

$\displaystyle\Gamma_{ab}^{f}(\widehat{n}_{a},\widehat{n}_{b})=\int\frac{{\rm
d}^{2}\widehat{n}}{4\pi}\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}\,f^{2}\sum_{A=+,\times}e^{A}_{ij}(\widehat{n})e^{A}_{kl}(\widehat{n}).$
(5.3)

Repeating the steps of section 4, we define
${\Gamma^{f-}_{ab}}=\Gamma_{ab}^{f}-\Gamma_{v}^{f}$, which transforms under
Lorentz boost as

$\delta_{K}{\Gamma^{f-}_{ab}}=2\bigg{(}(-v^{i}+\vec{v}\cdot\widehat{n}_{a}\widehat{n}_{a}^{i})\widehat{n}_{a}^{j}\widehat{n}_{b}^{l}\widehat{n}_{b}^{k}+(-v^{i}+\vec{v}\cdot\widehat{n}_{b}\widehat{n}_{b}^{i})\widehat{n}_{b}^{j}\widehat{n}_{a}^{k}\widehat{n}_{a}^{l}\bigg{)}\int\frac{{\rm
d}^{2}\widehat{n}}{4\pi}f^{2}\sum_{A=+,\times}e^{A}_{ij}(\widehat{n})e^{A}_{kl}(\widehat{n}),$
(5.4)

where the transformation of Eq.(3.8) has been used. Then, by using the fact
that, since $\Gamma_{ab}^{f}$ is quadratic in $\widehat{n}_{a}$, it is
invariant under $\widehat{n}_{a}\to-\widehat{n}_{a}$ or, equivalently, under
$\sigma_{ab}\to 1-\sigma_{ab}$, we get

$\delta_{K}{\Gamma^{f-}_{ab}}=\vec{v}\cdot(\widehat{n}_{a}+\widehat{n}_{b}){\Gamma^{f-}_{ab}}-\frac{1}{2}\left(\delta\widehat{n}_{a}\cdot\nabla_{{\widehat{n}_{a}}}{\Gamma^{f-}_{ab}}+\delta\widehat{n}_{b}\cdot\nabla_{{\widehat{n}_{b}}}{\Gamma^{f-}_{ab}}\right).$
(5.5)

Therefore, invariance under Lorentz boosts
($\delta_{K}{\Gamma^{f}_{ab}}^{-}=0$) leads to

$\vec{v}\cdot(\widehat{n}_{a}+\widehat{n}_{b})\bigg{(}2{\Gamma^{f-}_{ab}}-\sigma_{ab}\frac{{\rm
d}{\Gamma^{f-}_{ab}}}{{\rm d}\sigma_{ab}}\bigg{)}=0,$ (5.6)

which is solved by

$\displaystyle{\Gamma^{f-}_{ab}}=\alpha\,\sigma_{ab}^{2}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}\Gamma_{ab}^{f}=\alpha\,\sigma^{2}_{ab}+\Gamma_{v}^{f}.$ (5.7)

Implementing the $\sigma_{ab}\to 1-\sigma_{ab}$ symmetry of the overlap
function we find

$\displaystyle{\Gamma^{f}_{ab}}=\frac{\alpha}{2}\Big{(}\sigma_{ab}^{2}+(1-\sigma_{ab})^{2}\Big{)}+\Gamma_{v}^{f}.$
(5.8)

In addition, the identity

$\displaystyle\int{\rm
d}^{2}\widehat{n}_{a}\,\widehat{n}_{a}^{i}\widehat{n}_{a}^{i}=\frac{4\pi}{3}\delta^{ij},$
(5.9)

leads to the condition

$\displaystyle\int{\rm d}^{2}\widehat{n}_{a}\,\Gamma_{ab}^{f}=0,$ (5.10)

as the polarization tensors are traceless. Using Eq. (5.8) in Eq. (5.10) we
find that $\Gamma_{v}^{f}=-\alpha/3$. Therefore, the final expression for the
overlap function in the short-arm approximation turns out to be

$\Gamma^{f}_{ab}=\dfrac{\alpha}{2}\left(\sigma_{ab}^{2}+(1-\sigma_{ab})^{2}-\dfrac{2}{3}\right)=\dfrac{\alpha}{6}P_{2}(\widehat{n}_{a}\cdot\widehat{n}_{b})$,

(5.11)

with $P_{2}$ the $\ell=2$ Legendre polynomial $P_{\ell}$. It provides the same
value at opposite angles. The coefficient $\alpha$ is an arbitrary parameter,
which can be set to 3 to reproduce the value reported in the literature [11].

##  6 Conclusions

We have shown that the HD correlation function can be recovered by simple
symmetry arguments once it is realized that Lorentz symmetries act as
conformal symmetries on the celestial sphere and that the HD correlation
function has well-defined conformal transformation properties.

We believe that this result is useful to describe variations of the standard
HD function, for instance when there are additional polarization modes (vector
and/or scalar) that may arise in alternative-gravity theories [21, 22] for
which a different angular correlation functions are expected. Anisotropy,
linear, or circular polarization in the stochastic GW background [23] gives
rise to extra structure in the two-point correlation function and cannot be
written simply in terms of the angular separation of the two pulsars. One
relevant question we would like to investigate is if a deformation of the
conformal field theory on the celestial sphere can help in describing these
cases too.

Acknowledgments

We thank V. De Luca, A. Mitridate, K. Papadodimas, J. Romano, D. Racco and G.
Tasinato for useful discussions. A.R. is supported by the Boninchi Foundation
for the project “PBHs in the Era of GW Astronomy”.

##  Appendix A The logarithmic origin of the Hellings and Downs function

Logarithmic terms appear in the two-point function for conformal field
theories in two dimensions. However, we should also notice, that such terms
naturally appear in logarithimc conformal theories [24, 25]. The latter have
been employed in the description of percolation [26], the quantum Hall effect
[27], planar magnetohydrodynamics [28], AdS/CFT correspondence [29, 30]
whereas they have also been discussed in a cosmological context in Ref. [31].
These theories are conformal invariant in a broader sense. Usually, a generic
field $\phi(\vec{x})$ of dimension $w$ transforms under dilations as

$\displaystyle[D,\phi(\vec{x})]=\left(x^{i}\partial_{i}+w\right)\phi(\vec{x}).$
(A.1)

However, in the presence of second field $\Phi$ of the same dimension $w$, we
may have under dilations the transformation

$\displaystyle
i[D,\Phi_{\alpha}(x)]=\left(x^{i}\partial_{i}\delta_{\alpha}^{\beta}+\Delta_{\alpha}^{\beta}\right)\Phi_{\alpha}(x),~{}~{}~{}~{}a=1,2.$
(A.2)

where $\Phi_{\alpha}=(\phi,\Phi)$ is the logarithmic pair, and
$\Delta_{\alpha}^{\beta}$ is a $2\times 2$ matrix. In the standard case, where
$\Delta_{a}^{b}=w\,\delta_{a}^{b}$, the two-point correlator is

$\displaystyle\langle\Phi_{\alpha}(0)\Phi_{\beta}(\vec{x})\rangle=c\,\delta_{\alpha\beta}\,|\vec{x}|^{-2w}.$
(A.3)

However, in the most general case, the matrix $\Delta_{\alpha}^{\beta}$ need
not be diagonal, and it can be brought to its Jordan canonical form

${\bf\Delta}=\left(\begin{array}[]{cc}w&0\\\ 1&w\end{array}\right).$ (A.4)

Then, the two-point correlators
$G_{\alpha\beta}(\vec{x},\vec{y})=\langle\Phi_{\alpha}(\vec{x})\Phi_{\beta}(\vec{y})\rangle$
are found by solving the corresponding Ward identities, which are explicitly
written as [31]

$\displaystyle\left(2w+x\frac{\partial}{\partial
x}\right)G_{12}+G_{11}=0,\qquad\left(2w+x\frac{\partial}{\partial
x}\right)G_{11}=0,$ $\displaystyle\left(2w+x\frac{\partial}{\partial
x}\right)G_{22}+2G_{12}=0,\qquad\Delta_{\alpha\beta}G_{\beta\gamma}=G_{\alpha\beta}\Delta_{\gamma\beta},$
(A.5)

where $x=|\vec{x}|$. The solution of the above equations turns out to be

$\displaystyle G_{12}=-\frac{c}{2}{x^{2w}},~{}~{}~{}~{}~{}~{}~{}~{}G_{11}=0,$
$\displaystyle G_{22}=\frac{c}{x^{2w}}\left(\ln x+\frac{\beta}{c}\right).$
(A.6)

Therefore, for $w=-1/2$ and $G_{22}=\Gamma^{-}_{ab}$, we find that

$\displaystyle\Gamma_{ab}=c\,\sigma_{ab}\ln\sigma_{ab}+\beta\,\sigma_{ab}+\Gamma_{v},$
(A.7)

i.e., the functional form of the Hellings and Downs curve.

A similar treatment of the three-point function for fields at
$\vec{x}_{1},\vec{x}_{2}$ and $\vec{x}_{3}$ leads to the non-zero correlators
[31]

$\displaystyle
G_{122}=\langle\phi(\vec{x}_{1})\Phi(\vec{x}_{2})\Phi(\vec{x}_{3})\rangle=\frac{c_{1}}{2}\,x_{12}^{-w}x_{23}^{-w}x_{13}^{-w},$
(A.8) $\displaystyle
G_{222}=\langle\Phi(\vec{x}_{1})\Phi(\vec{x}_{2})\Phi(\vec{x}_{3})\rangle=c_{1}x_{12}^{-w}x_{23}^{-w}x_{13}^{-w}\Big{(}\ln(x_{12}x_{23}x_{13})+a\Big{)},$
(A.9)

where $x_{ij}=|\vec{x}_{i}-\vec{x}_{j}|$. Again for conformal dimension
$w=-1/2$, we get the three–point correlator of Eq. (4.46).

##  Appendix B The correlator for collinear pulsars

By using the sum helicity rule [32]

$\displaystyle\sum_{A=+,\times}\epsilon^{A}_{ij}(\widehat{n})\epsilon^{A}_{kl}(\widehat{n})=$
$\displaystyle\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl}-\delta_{ik}\widehat{n}_{j}\widehat{n}_{l}-\delta_{il}\widehat{n}_{j}\widehat{n}_{k}$
$\displaystyle-\delta_{jk}\widehat{n}_{i}\widehat{n}_{l}-\delta_{jl}\widehat{n}_{i}\widehat{n}_{k}+\delta_{ij}\widehat{n}_{k}\widehat{n}_{l}+\delta_{kl}\widehat{n}_{i}\widehat{n}_{j}+\widehat{n}_{i}\widehat{n}_{j}\widehat{n}_{k}\widehat{n}_{l},$
(B.1)

we find that

$\displaystyle\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}\widehat{n}_{a}^{k}\widehat{n}_{a}^{l}\bigg{(}\sum_{A=+,\times}\epsilon^{A}_{ij}(\widehat{n})\epsilon^{A}_{kl}(\widehat{n})\bigg{)}=(1-\widehat{n}\cdot\widehat{n}_{a})^{2}(1+\widehat{n}\cdot\widehat{n}_{a})^{2}.$
(B.2)

Then, it turns out that the two-point correlators
$C_{2}(0)=\Gamma_{ab}(\sigma)\Big{|}_{\sigma=0}$ for
$\widehat{n}_{b}=\widehat{n}_{a}$ and
$C_{2}(1)=\Gamma_{ab}(\sigma)\Big{|}_{\sigma=1}$ for
$\widehat{n}_{b}=-\widehat{n}_{a}$ are given by

$\displaystyle C_{2}(0)=\int\frac{{\rm
d}^{2}\widehat{n}}{4\pi}\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}}{(1-\widehat{n}\cdot\widehat{n}_{a})^{2}}\left(\sum_{A=+,\times}e^{A}_{ij}(\widehat{n})e^{A}_{kl}(\widehat{n})\right),$
$\displaystyle C_{2}(1)=\int\frac{{\rm
d}^{2}\widehat{n}}{4\pi}\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}\widehat{n}_{b}^{k}\widehat{n}_{b}^{l}}{1-(\widehat{n}\cdot\widehat{n}_{a})^{2}}\left(\sum_{A=+,\times}e^{A}_{ij}(\widehat{n})e^{A}_{kl}(\widehat{n})\right).$
(B.3)

By using Eq. (B.2), we find that

$\displaystyle C_{2}(0)$ $\displaystyle=\frac{1}{16\pi}\int{\rm
d}^{2}\widehat{n}\,\Big{(}1-\widehat{n}\cdot\widehat{n}_{a}\Big{)}^{2},$
$\displaystyle C_{2}(1)$ $\displaystyle=\frac{1}{16\pi}\int{\rm
d}^{2}\widehat{n}\,\Big{(}1-(\widehat{n}\cdot\widehat{n}_{a})^{2}\Big{)},$
(B.4)

from which the relation

$\displaystyle C_{2}(0)=2C_{2}(1)=\frac{1}{3}$ (B.5)

follows, reproducing the coefficients in Eq. (4.1).

##  Appendix C Proof that the integral of the HD correlation function over the
celestial sphere vanishes

In order to prove Eq. (4.33), let us fix $\widehat{n}_{b}=(0,0,1)$ and let
$\widehat{n}_{a}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. Then
from the definition of the two-point correlator in Eq. (4.9) we get

$\displaystyle\int{\rm d}^{2}\Omega_{2}\,\Gamma_{ab}=\int{\rm
d}^{2}\widehat{n}_{a}\,\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}\times\sum_{A=+,\times}\epsilon^{A}_{ij}(\widehat{n})\times(\mbox{terms
independent of $\widehat{n}_{a}$}).$ (C.1)

Now we want to evaluate the integral of the right-hand side of Eq. (C.1),
which due to symmetry should be written as

$\displaystyle\int{\rm
d}^{2}\widehat{n}_{a}\,\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}=\int{\rm
d}^{2}\widehat{n}_{a}\,\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-s\,\widehat{n}\cdot\widehat{n}_{a}}\Big{|}_{s=1}=A(s)n^{i}n^{j}+B(s)\delta^{ij}\Big{|}_{s=1},$
(C.2)

where $A,B$ are constants. Since the above expression is contracted with the
transverse and traceless polarization tensors $\epsilon^{A}_{ij}(\widehat{n})$
we have that in general

$\displaystyle\Big{(}A(s)n^{i}n^{j}+B(s)\delta^{ij}\Big{)}\epsilon^{A}_{ij}(\widehat{n})=0,$
(C.3)

and therefore,

$\displaystyle\int{\rm
d}^{2}\widehat{n}_{a}\,\frac{\widehat{n}_{a}^{i}\widehat{n}_{a}^{j}}{1-\widehat{n}\cdot\widehat{n}_{a}}\sum_{A=+,\times}\epsilon^{A}_{ij}(\widehat{n})=0~{}~{}~{}~{}{\rm
or}~{}~{}~{}~{}\int{\rm d}^{2}\Omega_{2}\,\Gamma_{ab}=0,$ (C.4)

as claimed. Let us note that we have inserted the parameter $s\geq 1$ in Eq.
(C.2) in order to avoid the pole at $\widehat{n}\cdot\widehat{n}_{a}=1$. We
could equally use a $+i\epsilon$ shift in the denominator. Both ways lead to
the vanishing of the integral in Eq. (C.4).

## References

  * [1] NANOGrav collaboration, G. Agazie et al., _The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background_ , _Astrophys. J. Lett._ 951 (2023) L8, [2306.16213].
  * [2] NANOGrav collaboration, G. Agazie et al., _The NANOGrav 15 yr Data Set: Observations and Timing of 68 Millisecond Pulsars_ , _Astrophys. J. Lett._ 951 (2023) L9, [2306.16217].
  * [3] EPTA, InPTA: collaboration, J. Antoniadis et al., _The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals_ , _Astron. Astrophys._ 678 (2023) A50, [2306.16214].
  * [4] EPTA collaboration, J. Antoniadis et al., _The second data release from the European Pulsar Timing Array - I. The dataset and timing analysis_ , _Astron. Astrophys._ 678 (2023) A48, [2306.16224].
  * [5] EPTA collaboration, J. Antoniadis et al., _The second data release from the European Pulsar Timing Array: V. Implications for massive black holes, dark matter and the early Universe_ , 2306.16227.
  * [6] D. J. Reardon et al., _Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array_ , _Astrophys. J. Lett._ 951 (2023) L6, [2306.16215].
  * [7] PPTA collaboration, A. Zic et al., _The parkes pulsar timing array third data release_ , 2306.16230.
  * [8] D. J. Reardon et al., _The Gravitational-wave Background Null Hypothesis: Characterizing Noise in Millisecond Pulsar Arrival Times with the Parkes Pulsar Timing Array_ , _Astrophys. J. Lett._ 951 (2023) L7, [2306.16229].
  * [9] H. Xu et al., _Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I_ , _Res. Astron. Astrophys._ 23 (2023) 075024, [2306.16216].
  * [10] R. W. Hellings and G. S. Downs, _Upper Limits on the Isotropic Gravitational Radiation Background from Pulsar Timing Analysis_ , _Astrophys. J. Lett._ 265 (1983) L39–L42.
  * [11] J. D. Romano and B. Allen, _Answers to frequently asked questions about the pulsar timing array Hellings and Downs correlation curve_ , 2308.05847.
  * [12] _Group Theory in Nonlinear Problems. Lectures Presented at the Nato Advanced Study Institute on Mathematical Physics Held in Istanbul, August 7-18, 1972_ , 1974.
  * [13] R. Penrose and W. Rindler, _Spinors and Space-Time_.  Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 4, 2011, 10.1017/CBO9780511564048.
  * [14] B. Oblak, _From the Lorentz Group to the Celestial Sphere_ , 8, 2015.  1508.00920.
  * [15] M. Maggiore, _Gravitational Waves. Vol. 2: Astrophysics and Cosmology_.  Oxford University Press, 3, 2018.
  * [16] A. I. Janis and E. T. Newman, _Structure of Gravitational Sources_ , _J. Math. Phys._ 6 (1965) 902–914.
  * [17] S. A. Teukolsky, _Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations_ , _Astrophys. J._ 185 (1973) 635–647.
  * [18] N. Bartolo, V. De Luca, G. Franciolini, A. Lewis, M. Peloso and A. Riotto, _Primordial Black Hole Dark Matter: LISA Serendipity_ , _Phys. Rev. Lett._ 122 (2019) 211301, [1810.12218].
  * [19] N. Bartolo, V. De Luca, G. Franciolini, M. Peloso, D. Racco and A. Riotto, _Testing primordial black holes as dark matter with LISA_ , _Phys. Rev. D_ 99 (2019) 103521, [1810.12224].
  * [20] C. Powell and G. Tasinato, _Probing a stationary non-Gaussian background of stochastic gravitational waves with pulsar timing arrays_ , _JCAP_ 01 (2020) 017, [1910.04758].
  * [21] Q. Liang and M. Trodden, _Detecting the stochastic gravitational wave background from massive gravity with pulsar timing arrays_ , _Phys. Rev. D_ 104 (2021) 084052, [2108.05344].
  * [22] Q. Liang, M.-X. Lin and M. Trodden, _A test of gravity with Pulsar Timing Arrays_ , _JCAP_ 11 (2023) 042, [2304.02640].
  * [23] J. R. Gair, J. D. Romano and S. R. Taylor, _Mapping gravitational-wave backgrounds of arbitrary polarisation using pulsar timing arrays_ , _Phys. Rev. D_ 92 (2015) 102003, [1506.08668].
  * [24] L. Rozansky and H. Saleur, _Quantum field theory for the multivariable Alexander-Conway polynomial_ , _Nucl. Phys. B_ 376 (1992) 461–509.
  * [25] V. Gurarie, _Logarithmic operators in conformal field theory_ , _Nucl. Phys. B_ 410 (1993) 535–549, [hep-th/9303160].
  * [26] H. Saleur, _Lectures on nonperturbative field theory and quantum impurity problems_ , cond-mat/9812110.
  * [27] V. Gurarie, M. Flohr and C. Nayak, _The Haldane-Rezayi quantum Hall state and conformal field theory_ , _Nucl. Phys. B_ 498 (1997) 513–538, [cond-mat/9701212].
  * [28] M. A. I. Flohr, _On fusion rules in logarithmic conformal field theories_ , _Int. J. Mod. Phys. A_ 12 (1997) 1943–1958, [hep-th/9605151].
  * [29] A. M. Ghezelbash, M. Khorrami and A. Aghamohammadi, _Logarithmic conformal field theories and AdS correspondence_ , _Int. J. Mod. Phys. A_ 14 (1999) 2581–2592, [hep-th/9807034].
  * [30] I. I. Kogan, _Singletons and logarithmic CFT in AdS / CFT correspondence_ , _Phys. Lett. B_ 458 (1999) 66–72, [hep-th/9903162].
  * [31] A. Kehagias and A. Riotto, _Operator Product Expansion of Inflationary Correlators and Conformal Symmetry of de Sitter_ , _Nucl. Phys. B_ 864 (2012) 492–529, [1205.1523].
  * [32] D. Anninos, V. De Luca, G. Franciolini, A. Kehagias and A. Riotto, _Cosmological Shapes of Higher-Spin Gravity_ , _JCAP_ 04 (2019) 045, [1902.01251].