11institutetext: Center for Informatics Research in Science & Scholarship
School of Information Sciences
University of Illinois at Urbana-Champaign, USA
11email<EMAIL_ADDRESS>
22institutetext: Research System Team, Microsoft, Redmond, USA
22email<EMAIL_ADDRESS>
# Unlocking Insights: Semantic Search in Jupyter Notebooks††thanks: Supported
by Microsoft Research.

Lan Li 11 0000-0003-4499-4126 Jinpeng Lv 22

###### Abstract

Semantic search, a process aimed at delivering highly relevant search results
by comprehending the searcher’s intent and the contextual meaning of terms
within a searchable dataspace, plays a pivotal role in information retrieval.
In this paper, we investigate the application of large language models to
enhance semantic search capabilities, specifically tailored for the domain of
Jupyter Notebooks. Our objective is to retrieve generated outputs, such as
figures or tables, associated functions and methods, and other pertinent
information.

We demonstrate a semantic search framework that achieves a comprehensive
semantic understanding of the entire notebook’s contents, enabling it to
effectively handle various types of user queries. Key components of this
framework include:

1). A data preprocessor is designed to handle diverse types of cells within
Jupyter Notebooks, encompassing both markdown and code cells.
2). An innovative methodology is devised to address token size limitations
that arise with code-type cells. We implement a finer-grained approach to data
input, transitioning from the cell level to the function level, effectively
resolving these issues.

###### Keywords:

semantic search large language model Jupyter Notebook

##  1 Introduction

In contrast to traditional keyword-based searches, which seek literal matches
or variations of query terms without full comprehension of their contextual
meaning, semantic search requires an understanding of the searcher’s intent
and the contextual meaning of terms within the searching space. Embedding
generated by language models has been applied to various areas, such as
finding relatedness among entities in Knowledge Graphs (KGs) [11], generating
KGs from semi-structured data, querying, and managing Knowledge Graph
Engineering (KGE) [12], and comparing whether records from a pair of
relational tables refer to the same entity or not [10]. All these applications
have demonstrated that embeddings can appropriately reflect the semantics for
different types of data, ranging from unstructured, semi-structured, to
structured datasets.

Jupyter Notebook empowers users to create narratives comprising live, re-
executable code, mathematical equations, descriptive text, interactive
dashboards, and various multimedia elements [7]. Therefore, a single Jupyter
Notebook, usually loaded in JSON format, is composed of various types of
information. With the widespread adoption of large language models (LLMs)
across diverse applications, yielding remarkable results, we extend an
invitation to LLMs to play a pivotal role in enhancing and empowering the
semantic search process within the domain of Jupyter Notebooks.

Based on our study on user requirements for searching Jupyter Notebooks within
the team, users are interested in exploring data visualizations, analyzing
plotted results, accessing tutorials for specific method usage, inspecting
code implementations of ad-hoc functions, and reviewing general output
information from the notebook contents. These requirements depend on a
comprehensive and nuanced semantic grasp of Jupyter Notebook. In a nutshell,
our assumption is that LLMs can comprehend and generate meaningful embeddings
for Jupyter Notebooks, thereby supporting the query process in a robust
manner.

In summary, this project is driven by several key goals, which include
exploring the potential of LLMs and addressing challenges from the following
perspectives:

  * •

Identifying Data Inputs for Jupyter Notebook content embeddings: pinpoint the
essential data inputs necessary for generating embeddings from Jupyter
Notebook content.

  * •

Resolving token size limitation restricted by embedding model: propose a
finer-grained granularity of data inputs (code-type cell) from cell level to
function level.

  * •

Studying code summaries by the latest GPT-4 model: study and confirm the
preservation of information when transitioning from the original code to code
summaries generated by the GPT-4 model.

  * •

Evaluating the ability of addressing users’ queries: accommodate various types
of user queries comprehensively.

##  2 Related Work: Transformers in Semantic Search and Code Summarization

Semantic search is composed of two tasks: 1. understanding the semantics of
queries and documents beyond keywords; 2. finding top k answers from a
document corpus given a query as search result [1]. With our case, semantic
search for Jupyter Notebooks, a JSON format data which is composed of both
textual part (i.e., markdown) and code part, requires a good understanding of
both types of contents.

Transformers are widely applied to Natural Language Process tasks, also shed
lights on semantic search area. Search applications like Google [2] and Bing
[3] employ transformers to provide semantically meaningful search results.
However, it’s important to note that these applications utilize limited BERT-
like encoder-only transformers. In contrast, Muennighoff et al. [1] introduce
a different approach by applying decoder-only transformers, referred to as
SGPT, for semantic search and the generating meaningful sentence embeddings.
The primary emphasis of the paper lies in evaluating SGPT’s performance on the
BEIR search benchmark, raising uncertainty about its effectiveness in
addressing various other semantic search tasks [4].

GPT-based models hold substantial promise for application in diverse domains,
encompassing education, medicine, history, mathematics, physics, and beyond.
These models can streamline various tasks, such as generating summaries,
answering questions, and offering personalized recommendations for users [5].
In particular, The foundation models pretrained on massive data and finetuned
to downstream tasks have been widely applied for code summarization, reducing
the burden of software development and maintenance [6].

Code summarization is the task of generating descriptions in natural language
to document the implementation details, data inputs, and data outputs, which
plays an important role in understanding the semantics of code snippets [6]
(See Table 1). In contrast to the automatic code summarization models
supported by CodeBERT and GPT-2, as discussed by [6], we leverage the state-
of-the-art GPT-4 111https://platform.openai.com/docs/models/gpt-4 model for
code summarization. GPT-4 model allows for a larger token size input, training
on more extensive datasets, and the ability to generate more complex
responses.

Table 1: One example code from the LeetCode website: the first solution from two-sum and code summary generated by _gpt-4-32k_ model. Code summary generated by the Language Model (LLM) is well-structured and provides in-depth explanations of its internal logic and mechanisms. Code |  ⬇ class Solution: def twoSum(self, nums: List[int], target: int) -> List[int]: seen = {} for i, value in enumerate(nums): #1 remaining = target - nums[i] #2 if remaining in seen: #3 return [i, seen[remaining]] #4 else: seen[value] = i #5
---|---
Code Summary |  The given code defines a class Solution with a method twoSum. This method takes a list of integers (numbers) and an integer target as input and returns a list of two indices from numbers that add up to the target. It uses a dictionary (seen) to store the numbers and their indices encountered so far, and calculates the remaining value needed to reach the target. If this remaining value is found in the dictionary, it returns the corresponding indices.

##  3 Approach and Demonstration

(a) This is the overall architecture, consisting of: 1). Embedding and
Completion models are used to generate embeddings for cells of each Jupyter
Notebook file; 2). Data insertion and database update: save the cell contents
and corresponding embeddings vectors into a vector database named Weaviate
[8], and update it every fifteen minutes; 3). Search engine: similarity search
and return top k related results.  (b) Our embedding generation pipeline
consists of three stages: Document chunking: Dividing the Jupyter Notebook
file into cells. Data preprocessing: Employing distinct preparation methods
for markdown and code cells. Requesting GPT models to generate embeddings.

Figure 1: Architecture of Semantic Search Framework (Left) and Pipeline of
Embedding Generation (Right)

We develop a semantic search framework for Jupyter Notebooks (see the
architecture on the left and the detailed pipeline of embedding generation
process on the right of Figure 1).

### Data Preprocessing

The most time-consuming and intricate aspect of our project lies in the
process of generating embeddings. To ensure the effectiveness of our semantic
search system for Jupyter Notebooks, we began by conducting a comprehensive
survey to gather user requirements regarding the search functionality. We
decide to focus on two fundamental types of cells within Jupyter Notebooks
accordingly: markdown cells, which typically contain descriptive overviews of
the notebook’s contents, and code cells, where implementation details are
documented.

Once we segmented the Jupyter Notebook files into these cells, various data
preprocessing methods are provided for each cell type. For markdown cells, our
preparators would remove various elements such as hyperlinks, special
characters (e.g, {,},:,“,”,’,!), and meaningless JSON field names. The well-
documented and explained markdown cells greatly enhances the semantic value of
the notebook’s content. However, we also recognized the diversity in
documentation styles among domain experts, data scientists, and engineers who
contribute to the Jupyter Notebook repository. In some cases, notebooks did
not contain markdown cells at all, but utilize separate files for
documentation purposes or instructions.

### Complete Embedding

In addressing these challenging situations, we recognized the necessity of
adapting our approach to generate embeddings exclusively from the code cells.
The primary challenge we encounter is by the token size limitation issues
imposed by our chosen embedding model, namely, _text-embedding-ada-002-v2_ ,
which allows for a maximum of 8191 tokens as input. There are several failed
practical use cases list as follows: 1). Long manual input of data values
within code cells. 2). Classes and functions are often consolidated into a
single cell in case of tutorial notebooks. 3). Functions are of intricate and
complex structures within individual cells.

To condense the code contents while retaining essential information, we switch
from creating embeddings at the cell level to a finer-grained approach that
focuses on individual functions. Firstly, we apply the Abstract Syntax Tree
(AST), an Python built-in module designed to facilitate the processing of
Python abstract syntax grammar trees. Our aim is to retain the core components
of the code, specifically class and function modules. However, if the token
size of the executed core code still exceeds our imposed limitations, we turn
to the latest GPT models to generate code summaries.

In a study conducted by Sun and Chen [9], various code summarization models,
including ChatGPT, were evaluated. ChatGPT, which is fine-tuned from a GPT-3.5
series model through reinforcement learning based on human feedback,
demonstrates noteworthy strengths. As highlighted by [9], ChatGPT exhibites an
enhanced ability to grasp the underlying meaning of the code, unaffected by
the code’s specific vocabulary. Additionally, ChatGPT tends to provide more
detailed descriptions of code behavior in the generated comments.

In our project, we leverage the capabilities of the cutting-edge GPT model,
specifically the _gpt-4-32k_ model, for code summarization. The _gpt-4-32k_
model accommodates input sizes of up to 32,000 tokens and excels at
understanding code semantics. We use a simple prompt to generate summaries:
“Generate a summary for the following code: \n $<code>$”.

### Data Storage and Queries

After finalizing the embedding generation process, data objects and their
corresponding embeddings are inserted into a vector database named Weaviate
(see Table 2). To ensure data consistency with the Jupyter Notebook
Repository, the database will be updated every 15 minutes. Subsequently, our
search engine accommodates various query types, including Exact Query (EQ),
User Defined Query (UDQ), and Code Summary Query (CSQ).

The first query type, EQ, is used to test whether the original cell can be
retrieved by inputting partial code or text copied and pasted from notebooks.
The second query type, UDQ, offers greater flexibility, allowing users to
express their requests and retrieve relevant content accordingly. Lastly, CSQ,
a form of ‘reverse engineering’, is employed to assess the system’s ability to
retrieve original code blocks with code summaries (in particular, to preserve
model consistency, we use the same _gpt-4-32k_ model to generate the code
summary).

These three query types represent distinct perspectives for understanding
notebooks, and we postulate that the search process for each query type serves
as a robustness test for large language models, guarantee their ability to
comprehend semantics across varying levels of complexity.

Due to data privacy considerations, we avoid disclosing specific contents
utilized in semantic search. Instead, we present a table depicting the
distances between query results generated by different query inputs and the
relevant notebooks (see Table 3). We figure out that the distance (calculated
by 1-cosine similarity) between the code summaries generated by the GPT model
and the actual content is remarkably close. This astonishing finding suggests
that GPT models might possess a deeper understanding of content semantics
compared to human searchers and even the original authors themselves!

Table 2: Schema Design notebook_id |  contents |  cell_type |  author_name |  modified_at |  created_at |  vector
---|---|---|---|---|---|---
“string” |  “text” |  “string” |  “string” |  “number” |  “number” |  “list of floats”
Notebook path |  cell contents |  [text, code] |  author name |  modification time |  creation time |  OpenAI Client Embedding
Table 3: Distances between Queries and Retrieved Contents Query Types | Perspective of Understanding | Distance
---|---|---
Exact Query | Author | 0.116
User Defined Query | Searcher | 0.163
Code Summary Query | GPT | 0.065

##  4 Conclusions and Future Work

In this study, we have delved into the potential of applying large language
models (LLMs) to semantic searches within Jupyter Notebooks. Our findings
confirm that these models exhibit a remarkable capability to comprehend both
textual and code-based data. They are adept at capturing the intricate
internal mechanisms and profound logic within the code, summarizing semantics
in a well-structured manner.

For future work, we have identified several key objectives:

  * •

Augmenting data inputs for LLMs: We plan to enrich the information provided to
LLMs by including additional details about cell contents, such as file paths,
cell types, and author names.

  * •

In addition to generating embeddings for Markdown-type and Code-type cells,
one potential task is to include outputs and graphs, which require a more
advanced multimodal LLMs.

  * •

Exploring code summarization at the cell level: We intend to investigate the
impact of applying code summarization to all cells without decomposing them
into function-level segments, aiming to assess its effect on search results.

  * •

Conducting qualitative and quantitative analyses: We will develop a
comprehensive set of both qualitative and quantitative analyses based on the
search results. These analyses will scrutinize how robust and adaptable
embeddings generated by LLMs are when dealing with codes of varying ’quality’.

All these analyses will contribute to understand and evaluate how robust and
adaptable LLM-generated embeddings are when dealing with codes of varying
“quality” and across different programming languages.

## References

  * [1] Muennighoff, N. (2022). Sgpt: Gpt sentence embeddings for semantic search. arXiv preprint arXiv:2202.08904.
  * [2] Nayak, P. (2019). Understanding searches better than ever before.
  * [3] Jeffrey Zhu, Mingqin Li, Jason Li, and Cassandra Oduola. (2021). Bing delivers more contextualized search using quantized transformer inference on NVIDIA GPUs in Azure.
  * [4] Koubaa, A., Boulila, W., Ghouti, L., Alzahem, A., & Latif, S. (2023). Exploring ChatGPT capabilities and limitations: A critical review of the nlp game changer.
  * [5] Liu, Y., Han, T., Ma, S., Zhang, J., Yang, Y., Tian, J., … & Ge, B. (2023). Summary of ChatGPT-Related Research and Perspective Towards the Future of Large Language Models. Meta-Radiology, 100017.
  * [6] Gu, J., Salza, P., & Gall, H. C. (2022, March). Assemble foundation models for automatic code summarization. In 2022 IEEE International Conference on Software Analysis, Evolution and Reengineering (SANER) (pp. 935-946). IEEE.
  * [7] Kluyver, T., Ragan-Kelley, B., Pérez, F., Granger, B. E., Bussonnier, M., Frederic, J., … & Willing, C. (2016). Jupyter Notebooks-a publishing format for reproducible computational workflows. Elpub, 2016, 87-90.
  * [8] Weaviate: an open-source vector database. (2023). Retrieved from https://weaviate.io/.
  * [9] Sun, W., Fang, C., You, Y., Miao, Y., Liu, Y., Li, Y., … & Chen, Z. (2023). Automatic Code Summarization via ChatGPT: How Far Are We?. arXiv preprint arXiv:2305.12865.
  * [10] Fang, L., Li, L., Liu, Y., Torvik, V. I., and Ludäscher, B. (2023). Kaer: A knowledge augmented pre-trained language model for entity resolution. Knowledge Augmented Methods for Natural Language Processing workshop in conjunction with AAAI 2023.
  * [11] Morales, C., Collarana, D., Vidal, M. E., & Auer, S. (2017). Matetee: A semantic similarity metric based on translation embeddings for knowledge graphs. In Web Engineering: 17th International Conference, ICWE 2017, Rome, Italy, June 5-8, 2017, Proceedings 17 (pp. 246-263). Springer International Publishing.
  * [12] Meyer, L. P., Stadler, C., Frey, J., Radtke, N., Junghanns, K., Meissner, R., … & Martin, M. (2023). Llm-assisted knowledge graph engineering: Experiments with chatgpt. arXiv preprint arXiv:2307.06917.


# On the energy flow of $\lambda$ in Hořava-Lifshitz cosmology

Ewa Czuchry<EMAIL_ADDRESS>National Centre for Nuclear Research,
Pasteura 7, Warsaw, Poland  Nils A. Nilsson<EMAIL_ADDRESS>SYRTE,
Observatoire de Paris-PSL, Sorbonne Université, CNRS UMR8630, LNE, 61 avenue
del’Observatoire, 75014 Paris, France

###### Abstract

Hořava-Lifshitz gravity has been proposed as a ghost-free quantum gravity
model candidate with an anisotropic UV-scaling between space and time. We
present here a cosmological background analysis of two different formulations
of the theory, with particular focus on the running of the parameter
$\lambda$. Using a large dataset consisting of Cosmic Microwave Background
data from Planck, Pantheon+ supernovae catalogue, SH0ES Cepheid variable
stars, Baryon acoustic oscillations (BAO), Cosmic Chronometers, and gamma-ray
bursts (GRB), we arrive at new bounds on the cosmological parameters, in
particular $\lambda$, which describes deviation from classical general
relativity. For the detailed balance scenario we arrive at the bound
$\lambda=1.02726\pm 0.00012$, and for beyond detailed balance the limit reads
$\lambda=0.9949^{+0.0045}_{-0.0046}$. We also study the influence of different
data sets and priors, and we find that removing low-redshift data generally
moves $\lambda$ closer towards UV values, whilst simultaneously widening the
error bars. In the detailed balance scenario, this effect is more noticeable,
and $\lambda$ takes on values that are significantly below unity, which
corresponds to the infrared limit of the theory.

##  I Introduction

General Relativity (GR), one of the most successful physical theories, is
widely known to be non-renormalizable. Therefore, its application to very
small distances and high energies, such as at the very early Universe is
expected to be inconsistent. In light of this, a different theory of gravity
was recently proposed by Hořava [1] in order to capture the quantum effects of
the gravitational field in the early stages of the Universe. This theory is a
fascinating proposal of modified gravity equipped with an anisotropic scaling
at the Planck scale given in term of a critical Lifshitz exponent which
contributed to the name Hořava-Lifshitz (HL) gravity. This Lifshitz scaling in
the ultraviolet (UV) regime inevitably breaks Lorentz invariance
explicitly111See for example [2] for a discussion of explicit and spontaneous
symmetry-breaking in gravity..

By giving up on the idea of space-time invariance under four-dimensional
diffeomorphisms it becomes possible to add higher spatial-derivative terms to
the Lagrangian in a way that makes the theory renormalizable at high energies.
The classical formulation by Hořava [1] included in the potential part of the
action only terms which may be derived from a superpotential (the so called
”detailed balance condition”), thus limiting the number of its terms to those
quadratic in curvature and with independent couplings. Subsequent
formulations, the so called Sotiriou-Visser-Weinfurtner (SVW) generalisation
[3] included, relaxed this condition and included cubic terms in curvature in
the gravitational action.

The main feature distinguishing HL from GR is the explicit lack of invariance
under four-dimensional diffeomorphisms; in HL, there exists an ultraviolet
fixed point, where time and space scale differently, and so HL is naturally
expressed using the Arnowitt-Deser-Misner (ADM) 3+1 formulation [4]. The most
straightforward option for a group which can accommodate this property is the
set of foliation-preserving diffeomorphisms, which includes time
reparametrizations and three-dimensional spatial diffeomorphisms. Under this
symmetry group, the kinetic term of the action acquires an additional coupling
denoted by $\lambda$. In GR there is no need for that coupling as the very
specific linear combination of curvature terms contained within the classical
Hilbert-Einstein action remains unchanged under general four-dimensional
diffeomorphisms. In HL theory the limit $\lambda\rightarrow 1$ is supposed to
recover classical general relativity. This should in theory provide its low-
energy limit in a straightforward manner, but achieving this limit presents
some issues: on one hand, the critical Lifshitz exponent $z$ and the
$(3+1)$-foliation parameter $\lambda$ are the parameters of HL theory, where
$\lambda$ is also associated with a restricted foliation corresponding to
$z=1$ in the Lifshitz scaling. As $z$ approaches unity in the low-energy
limit, it is also necessary that $\lambda\rightarrow 1$ and consequently that
Lorentz invariance is restored. Additionally, the familiar ADM foliation of GR
is also recovered. Because the coupling constant $\lambda$ disrupts the
general covariance in the presence of space-time diffeomorphisms, it is
thought that values of $\lambda$ different from $1$ will cause the model to
deviate from GR. On the other hand, due to the reduced symmetry, the action of
HL gravity [1] differs from the classical Einstein-Hilbert action even in the
infrared limit, where only terms up to second order in spatial derivatives are
considered. Even though the IR limit differs from GR, it is supposed that one
can still obtain it as an approximate solution at long distances; however, it
seems that this connection with GR is problematic, since in order to obtain
it, it is necessary to disregard the higher order derivative terms in the
action. Additionally, the GR limit presents issues because HL contains an
extra degree of freedom present as a scalar mode, which causes a significant
challenge in terms of the credibility of the theory [3, 5, 6].

In recent years, significant work has been done on HL gravity and its
cosmological aspects (see [7] for an extensive review), including several
early extensions attempting to cure some of its initial problems [8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 3], as well as studies of black holes
[21], dark energy [22], the black-hole shadow [23, 24], gravitational waves
[25, 26], and many more (see [7]). There has been also a lot of discussion on
its problems and inconsistencies (see e.g. [3, 9, 27]) such as the existence
of a parity violating term [3], ghost instabilities and problems with strong
coupling at very low energies [28, 29], wrong sign and very large value of the
cosmological constant [30, 31], problems with power counting renormalization
of the scalar mode propagation [32, 12]. Some authors state that most of the
shortcomings can be cured by adding additional terms to the superpotential, or
relaxing the projectability condition which assumes that the lapse function
$N$ depends only on time ${N}={N}(t)$, resulting in its non-projectable
version. Additionally, given that the HL theory is an extension of GR in the
high energy regime, it is unsurprising that also its quantum gravity aspects
have sparked considerable interest [33, 34, 35, 36, 37, 38, 39].

In the first ”projectable” versions (${N}={N}(t)$), including the detailed
balance formulation (and further the Sotiriou-Visser-Weinfurtner extension [3]
with the relaxing of the detailed balance condition), application of the
action to the maximally symmetric Friedmann-Lemaitre-Robertson-Walker (FLRW)
metrics results in the instability of the extra scalar mode. In general,
$\lambda$ runs with energy and has the three IR fixed points: $\lambda=1/3$,
$\lambda=1$ and $\lambda=\infty$, and the unstable quantum behaviour of the
scalar mode results in a ghost, appearing in the range $1/3<\lambda<1$,
whereas it is only classically unstable in other ranges: $\lambda<1/3$ or
$\lambda>1$ [3, 5, 6]. The classical instability in the UV range, when we
expect quantum effects, is rather of lesser importance.

The ”non-projectable” version of HL, the so called called ”healthy extension”
[9] was supposed to eliminate this unphysical scalar degree of freedom;
nevertheless, there has been a controversy regarding the actual health
benefits of these extensions, as was demonstrated in [40]. There, it was shown
that difficulties with strong coupling appear in the low-energy limit, and
that the limit itself is problematic, since cubic terms in the action diverge
when $\lambda\rightarrow 1$. As a rebuttal, the authors of the original paper
have put forth the argument that the strong coupling scale could potentially
surpass the cutoff point for derivative expansions [41], which would introduce
a substantially lower scale of Lorentz violation.

Besides theoretical considerations there have been several attempts to put
observational bounds on $\lambda$ in addition to other HL gravity parameters.
The first works on observational bounds on HL gravity parameters such as [15]
and our previous paper [42] has manually set $\lambda$ to unity in the
analysis by considering the data to be too far in the IR compared to the scale
where this parameter can be expected to run. In [43] and the work by one us
[44], the authors considered a constant $\lambda$ and reabsorbed it into other
parameters. It is however worth mentioning that [16] obtained the first
constraints on $\lambda$ from cosmological data, reporting
$|\lambda-1|\lesssim 0.02$, while one of us [45] obtained $0.95\leq\lambda\leq
1.16$ by exploiting bounds on the Hubble parameter tension, and [46]
investigated the effects of a $\lambda$ phase transition in AdS black holes.

There are also other studies [17] providing bounds on the different parameters
of HL cosmology based on the so-called healthy extension of HL [9], where the
authors employ an effective-field theory and also consider linear
perturbations around the background. This study is however based on a flat
model, whereas there is ongoing discussion on the possible non-zero value of
the curvature parameter both in $\Lambda$CDM model [47] as its alternatives
including HL cosmology.

Therefore, a natural question arises: how does the cosmological behaviour of
Hořava-Lifshitz gravity change when $\lambda$ is allowed to vary, since it is
supposed to control the breaking of Lorentz invariance, and it might flow to
unity deep in the IR. When we allow $\lambda$ to vary we can set bounds on the
“level“ of Lorentz violation at the energy scales available to us through
data. This was done in [16] but this analysis simply took into account all the
available then data and did not investigate the influence of different energy
scales on the values of $\lambda$, which is what we do in the present paper.
Moreover, in this manuscript we use a larger and more updated dataset with an
additional number of high-energy sources such as gamma-ray bursts, which may
have big influence on numerical results.

In this paper we analyze the influence of including/excluding high/low energy
observational data on the values of $\lambda$ and perform a detailed
examination regarding how removing low-redshift data impact the obtained
bounds on this parameter. An intriguing matter comes to light, whether higher
redshift data could push $\lambda$ further from the classical limit at the
value of unity, as would be expected of HL or similar extensions of Einstein
gravity (with additional terms in the action). In this paper we perform the
analysis of including different data sets and priors on the value of $\lambda$
in two projectable versions of HL, one with imposed detailed balance condition
and one with this condition relaxed.

This paper is organized as follows: We first give a brief overview of the
theory of HL cosmology in both scenarios under consideration. We then go on to
describe how we rewrite the equations to allow for numerical analysis.
Finally, we present our results and discuss them.

##  II Hořava-Lifshitz cosmology

Here we briefly outline the equations governing cosmological evolution in
Hořava-Lifshitz gravity [11, 18]. For a more in-depth discussion we refer to
our recent paper [42], and also to [7].

Hořava-Lifshitz gravity offers the possibility of creating a theory of gravity
which remains finite and well defined at high energies, while also reproducing
GR in the classical regime. This is achieved by introducing a fixed point in
the UV region of the renormalization group, where time and space behave
differently under scale transformations. Specifically, the scaling relations
at the Planck scale should satisfy $t\,\rightarrow\,b^{-z}t$,
$x^{i}\,\rightarrow\,b^{-1}x^{i}$, $i=1,2,3$, where $b$ is a scaling parameter
and $z$ is the critical Lifshitz exponent which characterizes the fixed point.
As usual, $t$ stands for time and $x^{i}$ for spatial coordinates. Different
models can be identified by specific choices of $z$; for a pure gravity theory
in $d$ spatial dimensions which is invariant under foliation-preserving
diffeomorphisms and is power-counting renormalizable, $z$ must be greater than
or equal to $d$, equivalently in 4-dimensional physical space-time to $z\geq
4$ [7]. In order to restore Lorentz invariance, $z$ needs to be set to unity.

Due to the anisotropic scaling present in the theory, it is useful to write
down the metric using the ADM decomposition:

$ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (1)

where $N$ and $N^{j}$ are the lapse function and shift vector, respectively,
and $g_{ij}$ is the spatial metric ($i,j=1,2,3$). Given this, we can express
the most general form of the theory as:

$S=\int d^{3}xdtN\sqrt{g}\left[K^{ij}K_{ij}-\lambda
K^{2}-\mathcal{V}(g_{ij})\right],$ (2)

where $g$ is the determinant of the spatial metric, $\lambda$ is the running
coupling and $\mathcal{V}$ is a potential. $K_{ij}$ represents the extrinsic
curvature. The potential term contains only dimension 4 and 6 operators which
can be constructed from the spatial metric $g_{ij}$. The square $K^{ij}K_{ij}$
and its trace-squared $K^{2}$ are individually invariant under the reduced
symmetry group, however for $\lambda=1$ the full kinetic term
$K^{ij}K_{ij}-K^{2}$ acquires invariance under four-diffeomorphisms.

###  II.1 Detailed balance (DB)

The detailed-balance condition is a way to reduce the number of terms in the
action (2) by assuming that it should be possible to derive $\mathcal{V}$ from
a superpotential W [48, 30]:

$\mathcal{V}=E^{ij}\mathcal{G}_{ijkl}E^{kl},\quad
E^{ij}=\frac{1}{\sqrt{g}}\frac{\delta W}{\delta
g_{ij}},\quad\mathcal{G}^{ijkl}=\frac{1}{2}\left(g^{ik}g^{jl}+g^{il}g^{jk}\right)-\lambda
g^{ij}g^{kl}.$ (3)

which for $\lambda=1$ reduces to the standard Wheeler-DeWitt metric. From this
the most general action can be written as:

$\displaystyle S_{db}=\int\text{d}^{3}x\,\text{d}t\sqrt{g}N$
$\displaystyle\Bigg{[}\frac{2}{\kappa^{2}}\left(K_{ij}K^{ij}-\lambda
K^{2}\right)+\frac{\kappa^{2}}{2\omega^{4}}C_{ij}C^{ij}-\frac{\kappa^{2}\mu}{2\omega^{2}}\frac{\epsilon^{ijk}}{\sqrt{g}}R_{il}\nabla_{j}R^{l}_{k}$
(4)
$\displaystyle+\frac{\kappa^{2}\mu^{2}}{8}R_{ij}R^{ij}+\frac{\kappa^{2}\mu^{2}}{8(1-3\lambda)}\left(\frac{1-4\lambda}{4}R^{2}+\Lambda
R-3\Lambda^{2}\right)\Bigg{]},$ (5)

where $C^{ij}$ is the Cotton tensor, $\epsilon^{ijk}$ is the totally
antisymmetric tensor, and the parameters $\kappa$, $\omega$, and $\mu$ have
mass dimension $-1$,$0$, and $1$, respectively. This action has been obtained
from (2) by analytic continuation of the parameters $\mu\mapsto i\mu$ and
$\omega^{2}\mapsto-i\omega^{2}$, which enables positive values of the bare
cosmological constant $\Lambda$, which does not happen in the original
formulation.

As mentioned in the Introduction, the coupling constant $\lambda$ runs with
energy in a complicated manner and may reach one of the IR fixed points
$\\{1/3;1;\infty\\}$ [48]. From the point of view of scalar perturbations it
was pointed out in [3] that the range $1/3<\lambda<1$ leads to instabilities.
Unfortunately, also the flow interval of $\lambda$ between the UV and IR
regimes exhibits an unstable quantum behaviour of the scalar mode [3, 5, 6].
An effort was made to address this problem by proposing, as suggested in [5],
the enforcement of an analytic continuation of a particular form; however,
upon performing the analytic continuation as described in [5], it becomes
evident that the desirable UV behaviour is compromised and consequently,
instabilities resurface at high energies.

If we exclude the use of analytic continuation, the only relevant scenario
which permits a potential flow towards General Relativity (at $\lambda=1$) is
when $\lambda\geq 1$, as the region for values of $\lambda$ less than or equal
to 1/3 is separated by a problematic interval. However, in this paper we _have
not imposed this condition on $\lambda$_. Rather, we left it as free as
possible, the idea being that certain formulations of HL gravity may be ruled
out by the model preferring unphysical values of $\lambda$ when confronted
with data.222In the initial numerical results, we noted that whilst $\lambda$
does indeed take on values smaller than unity, it never gets close to $1/3$
(which is parameter singularity), and we therefore reintroduced the
$\lambda>1/3$ prior to save computation time, without loss of generality. Also
the idea was to fit the observational data without a priori assumption,
allowing further theoretical considerations based on obtained values. For
completeness, we also investigate the case when $\lambda\geq 1$.

Other formulations other than the one presented above exist; for example, one
can improve the low-energy behaviour of the theory by means of an IR-
modification parameter which for example allows for a consistent Minkowski
vacuum, as was done in [49, 22]. Such an IR-modification constitutes a so-
called soft breaking of the detailed-balance condition, and we do indeed see
these features when breaking this condition in this paper. Modifications can
also be introduced in the UV limit in the form of fourth, fifth, and sixth
order derivative terms; a UV detailed-balance condition is normally in place
which cancels these terms, but one may break this condition softly, and this
is in fact necessary in order to obtain scale-invariant scalar perturbations,
as was shown in [50, 51]. In this paper, we use the formulation from [1] and
[29] which possesses desirable features when relaxing the detailed-balance
condition; for completeness, we also study the case when this condition is in
place.

We populate our model with the canonical matter and radiation fields
represented by the energy densities (and pressures) $\rho_{m}$ ($p_{m}$) and
$\rho_{r}$ ($p_{r}$), both of which are subject to the continuity equation
$\dot{\rho}+3H(\rho+p)=0$. Here, an overdot represents a time derivative.
Moreover, we use the projectability condition [48] $N=N(t)$, and we use the
standard FLRW line element $g_{ij}=a(t)^{2}\gamma_{ij}$, $N^{i}=0$, where
$\gamma_{ij}$ is the maximally symmetric constant curvature metric. It is
important to remember that in theories which violate Lorentz symmetry the
gravitational constant appearing in the gravitational action $G_{\text{grav}}$
generally does _not_ coincide with the one which appears in the Friedmann
equations, $G_{\text{cosmo}}$ [52, 16]. This could in principle be used to set
bounds on $\lambda$ (as was done in [45]), but we will not adopt this approach
in this paper.

Varying the action (4) w.r.t $N$ and $a$ we arrive at the Friedmann equations
for the detailed balance scenario:

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\kappa^{2}}{6(3\lambda-1)}\left[\rho_{m}+\rho_{r}\right]+\frac{\kappa^{2}}{6(3\lambda-1)}\left[\frac{3\kappa^{2}\mu^{2}K^{2}}{8(3\lambda-1)a^{4}}+\frac{3\kappa^{2}\mu^{2}\Lambda^{2}}{8(3\lambda-1)}\right]-\frac{\kappa^{4}\mu^{2}\Lambda
K}{8(3\lambda-1)^{2}a^{2}},$ (6)
$\frac{\text{d}}{\text{d}t}\frac{\dot{a}}{a}+\frac{3}{2}\left(\frac{\dot{a}}{a}\right)^{2}=-\frac{\kappa^{2}}{4(3\lambda-1)}[p_{m}+p_{r}]-\frac{\kappa^{2}}{4(3\lambda-1)}\left[\frac{\kappa^{2}\mu^{2}K^{2}}{8(3\lambda-1)a^{4}}-\frac{3\kappa^{2}\mu^{2}\Lambda^{2}}{8(3\lambda-1)}\right]-\frac{\kappa^{4}\mu^{2}\Lambda
K}{16(3\lambda-1)^{2}a^{2}}.$ (7)

We can therefore define $G_{\text{cosmo}}=\kappa^{2}/(3\lambda-1)$ and
$\kappa^{4}\mu^{2}\Lambda=8(3\lambda-1)^{2}$ by requiring that (6,7) coincide
with the standard Friedmann equations. Clearly, when Lorentz invariance is
restored, $\lambda$ is set to unity and $G_{\text{grav}}=G_{\text{cosmo}}$.
Under detailed balance, and using the units $8\pi G_{\text{grav}}=1$, we are
lead to $\kappa^{2}=4$, $\mu^{2}\Lambda=2$, and by introducing the standard
density parameters we arrive at the Friedmann equation suitable for our
analysis:

$H^{2}=H_{0}^{2}\left[\frac{2}{3\lambda-1}\Large(\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z^{4})\Large)+\Omega_{K0}(1+z)^{2}+\omega+\frac{\Omega_{K0}^{2}}{4\omega}(1+z)^{4}\right],$
(8)

where $H$ denotes the Hubble parameter and the subscript $0$ indicates the
value as measured today. We have also introduced a parameter
$\omega=\Lambda/(2H_{0}^{2})$333Which is not to be confused with the IR-
modification parameter, which is usually also denoted by $\omega$.. A
characteristic feature of HL theory is the appearance of a dark radiation term
in (8): $\Omega_{k0}^{2}/4\omega$, and we can express this in terms of the
effective number of neutrino species $\Delta N_{\rm eff}$ present during the
BBN epoque (See [15, 16, 42]) as $\Omega_{k0}^{2}/4\omega=0.13424\Delta N_{\rm
eff}\Omega_{r0}$. We can also obtain a constraint from the $z=0$ limit, where
$H|_{z=0}=H_{0}$, which reads:
$(1-\Omega_{k0}-\omega-\Omega_{k0}^{2}/(4\omega))(3\lambda-1)/2=\Omega_{m0}+\Omega_{r0}$.
We abbreviate the Detailed Balance case as DB.

###  II.2 Beyond detailed balance (BDB)

There has been an ongoing discussion in the literature whether the detailed
balance condition is too restrictive [11, 18, 7], or if relaxing it can cure
the theory from the quantum instabilities of the scalar mode [9, 6, 53, 54,
55]. On one hand the detailed-balance condition introduces a superpotential,
which might simplify the quantization process significantly; on the other
hand, this condition is not fundamental, but helps in e.g., reducing the
number of independent couplings in the theory.

Therefore, we may safely consider the Sotiriou-Visser-Weinfurtner (SVW) [3]
generalization with this condition relaxed, after which the potential
$\mathcal{V}$ contains additional terms. In this scenario the action includes
quantities not only up to quadratic in curvature, as in the original HL
formulation, but also cubic ones, while suppressing parity violating terms.
Applying the generalized action to the maximally symmetric constant curvature
metric results in the following analogue of the Friedmann equations [29, 27]:

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{2\sigma_{0}}{3\lambda-1}(\rho_{m}+\rho_{r})+\frac{2}{3\lambda-1}\left[\frac{\Lambda}{2}+\frac{\sigma_{3}K^{2}}{6a^{4}}+\frac{\sigma_{4}K}{6a^{6}}\right]+\frac{\sigma_{2}}{3(3\lambda-1)}\frac{K}{a^{2}},$
(9)
$\frac{\text{d}}{\text{d}t}\frac{\dot{a}}{a}+\frac{3}{2}\left(\frac{\dot{a}}{a}\right)^{2}=-\frac{3\sigma_{0}}{3\lambda-1}\frac{\rho_{r}}{3}-\frac{3}{3\lambda-1}\left[-\frac{\Lambda}{2}+\frac{\sigma_{3}K^{2}}{18a^{4}}+\frac{\sigma_{4}K}{6a^{6}}\right]+\frac{\sigma_{2}}{6(3\lambda-1)}\frac{K}{a^{2}},$
(10)

where $\sigma_{i}$ are arbitrary constants. As in the detailed balance
scenario we find $G_{\text{cosmo}}=6\sigma_{0}/(8\pi(3\lambda-1))$, where
$\sigma_{0}=\kappa^{2}/12$. Using the same procedure as for detailed balance
in units where $8\pi G_{\text{grav}}=1$ we rewrite (9) to read:

$H^{2}=H_{0}^{2}=\left[\frac{2}{3\lambda-1}\Big{(}\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z)^{4}+\omega_{1}+\omega_{3}(1+z)^{4}+\omega_{4}(1+z)^{6}\Big{]}+\Omega_{k0}(1+z)^{2}\right],$
(11)

where we have, for convenience, introduced the following dimensionless
parameters, $\omega_{1}=\sigma_{1}/(6H_{0}^{2})$,
$\omega_{3}=\sigma_{3}H_{0}^{2}\Omega_{k0}^{2}/6$,
$\omega_{4}=-\sigma_{4}\Omega_{k0}/6$. Additionally, we impose $\omega_{4}>0$
in order for the Hubble parameter to be real for all $z$. Moreover, we can
extract a constraint from the $z=0$ limit of (11), which then reads
$(1-\Omega_{k0})(3\lambda-1/2=\Omega_{m0}+\Omega_{r0}+\omega_{1}+\omega_{3}+\omega_{4}$.
Much like for detailed balance, we can eliminate another parameter through
constraints by considering that the $\omega_{4}$ term corresponds to a
quintessence-like kinetic field. Therefore it is shown in [15] and references
therein that we get the following constraint at the time of BBN:
$\omega_{3}=0.13424\Delta N_{\rm eff}\Omega_{r0}-\omega_{4}(1+z_{BBN})^{2}$.
Here $z_{BBN}\approx 4\cdot 10^{4}$ is the redshift at BBN. We abbreviate the
Beyond Detailed Balance case as BDB.

##  III Constraints from cosmological data

###  III.1 The data

In order to carry out the parameter estimation we used a Markov-Chain Monte
Carlo (MCMC) method with a large cosmological data set. Since we are
interested in the running of $\lambda$ with energy, we include here a short
discussion of the different energy levels of the cosmological probes we use.
It is not a priori clear whether (or how) the running of $\lambda$ depends on
this energy.

Starting at high redshift (early times), our probes are the Cosmic Microwave
Background (CMB) and Baryon Acoustic Oscillations (BAO). The CMB originates
from the surface of last scattering at redshift $z_{\star}\sim 1100$, when the
temperature of the Universe was around $T\geq 3000$K, corresponding to an
energy of around $0.26$eV. This is complemented by BAO observations, which
also originate from recombination and manifest as acoustic peaks in the CMB
power spectrum; however, they also affect the distribution of local galaxies
and are therefore sensitive to different parameters compared to the CMB whilst
still being an early Universe probe.

The astrophysical sources we used have different properties, in that they
originate in the _late_ Universe, and whilst the emission energies are in
general higher than CMB and BAO, they do not offer any information about the
state of the Universe at early times. Supernovae Type 1a are used as standard
rulers and outputs more energy than the rest of its host galaxy in a very
short time. Their energy is less important in this context as they are used as
distance rulers by fitting their spectra; however, they are certainly high-
energy events, emitting neutrinos with energies up to $\sim 40$ MeV having
been detected [56]. The Cosmic Chronometer (CC) data set is based on passively
evolving galaxies, and we only sample the expansion history with these
objects. Last and most energetic, we also have gamma-ray bursts, which are the
most violent explosions in the known Universe; photons from gamma-ray bursts
have been found to reach energies up to $96$ GeV [57]. For all details about
the data sets and their implementation, see Appendix A and references therein.

###  III.2 Results

In order to derive constraints on the parameters we used (8) and its
associated constraint equations in our MCMC analysis. As we are mainly
interested in the running of $\lambda$ we present here only the marginalized
posteriors of this parameter. We include all parameter fits in Table 1 and 2
which can be found in Appendix B, where we are specifically interested in
$\Omega_{k0}$ and $\Delta N_{\rm eff}$, since previous work ([15, 16, 42])
suggests that it is possible to differentiate between the two scenarios using
these quantities.

In our analysis we find that in the detailed-balance scenario using all
available data, $\lambda$ takes the value $\lambda=1.02726\pm 0.00012$ at
$1\sigma$ confidence level, when not imposing the hard prior $\lambda\geq 1$;
its marginalized posterior is shown in Figure 1(b)444The MCMC chains which
generated the posteriors passed the standard convergence criteria detailed in
[58].. After imposing the prior on $\lambda$, we instead obtain
$\lambda=1.046\pm 0.0023$, and note that this value is a few percent larger,
but the $1\sigma$ error bars are one order of magnitude larger, which can be
seen in Figure 1(a). In order to investigate the possible running of $\lambda$
with energy, we carry out the same analysis by systematically removing data in
two steps: in the first step, we remove the Hubble parameter measurements from
Cosmic Chronometers (CC), and in the second step we remove everything except
the truly early-Universe probes, CMB and BAO. For $\lambda$, the results are
displayed in Figure 1, and we observe that the results are different depending
on whether we impose the Detailed-Balance condition or not. First of all,
$\lambda$ generally seems to take on higher values under Detailed Balance; for
this case, removing CC data pushes $\lambda$ to take values close to unity,
but further removing data moves it back up toward higher values, but with much
larger error bars. For both DB and BDB, the CMB+BAO combination seems to give
the largest errors by far; this is especially pronounced in DB (no prior) and
BDB (prior).

The case with the fewest assumptions and priors is the BDB (no prior) case
shown in Figure 1(d), and the situation here is somewhat reversed compared to
the other three; here, removing CC data produces a higher value of $\lambda$
than with the full dataset, which is in contrast to our other results. Also,
the case with all data overlaps significantly with that of CMB+BAO, both of
which have means firmly in the $\lambda<1$ region. In almost all presented
scenarios, in DB and BDB with the prior $\lambda\geq 1$ imposed and in DB with
no prior we observe that removing Hubble data from CC strongly pushes
$\lambda$ close to its IR limit. Only in the BDB scenario with no imposed
prior does removing that data result in $\lambda$ changing sign and increasing
$|\lambda-1|$. Furthermore, removing Supernovae Type 1a pushes $\lambda$ even
further into UV, except in the BDB with no prior scenario.

Another interesting quantity to analyse is the difference between $G_{\rm
grav}$ and $G_{\rm cosmo}$ which can be inferred directly from $\lambda$
through the quantity $|G_{\rm cosmo}/G_{\rm grav}-1|$. Here, we find that for
detailed balance, the difference between local $G$ and cosmological $G$ is in
general smaller than $\sim 5.8\%$, the largest discrepancy being in the case
of all data when including a prior on $\lambda$, where we find $|G_{\rm
cosmo}/G_{\rm grav}-1|=0.0574\pm 0.0030$. For beyond detailed balance, this
quantity takes on smaller values, generally $<2\%$, which is of course
directly linked to the values obtained on $\lambda$.

As in our previous work [42] we find here that $\Omega_{k0}$ is distinctly
non-zero in the Detailed Balance formulation of HL cosmology, which is the
same as was found in [44, 43]. We also briefly point out our results on the
Hubble constant, which are somewhat different from those of previous
background analysis [42, 44], where values close to $h=0.71$ were found for
the case of all data, which eased the Hubble parameter tension. In our present
results, we find values closer to that of $\Lambda$CDM, with $h=0.6488\pm
0.0012$ for DB and $h=0.6813\pm 0.048$ for BDB, both with all data; the BDB
value closely resembles that obtained by Planck ($h=0.6844\pm 0.091$) using
TE+lowE spectra whilst assuming a base $\Lambda$CDM model. On the other hand,
the DB case takes values close to the Planck TT+TE+EE+lowE+lensing results
when considering a $\Lambda$CDM model extended by allowing for a non-zero
$\Delta N_{\rm eff}$ [59]. A full analysis of the Hubble tension lies beyond
the scope of this paper.

(a) DB, prior

(b) DB, no prior

(c) BDB, prior

(d) BDB, no prior

Figure 1: Normalized posterior distribution functions for $\lambda$. Here,
SNe1a includes Cepheid-calibrated supernovae.

Another interesting result is the value of $\Delta N_{\rm eff}$, which takes
on values greater than $0.2$ in almost all cases, and a $1\sigma$ upper limit
of $\Delta N_{\rm eff}\leq 0.75$ for BDB (prior, all data). As this is
slightly surprising it is worth reiterating that this chain passed all
convergence criteria detailed in [58]. Therefore, with all other parameters
taking on reasonable values (and similar results found in [42, 44]) one may
have to entertain the possibility of a fourth neutrino species present in HL
cosmology. Indeed, it was recently suggested that a fourth neutrino might
solve the $H_{0}$ tension problem [60]; in this paper, the authors arrive at a
value of $N_{\text{eff}}\approx 4$ (effective number of neutrino species),
which is far higher than our results. Moreover, our results fall outside BBN
limits as reported in [61, 62] ($-1.7\leq\Delta N_{\rm eff}\leq 2.0$), but
seem to agree somewhat with limits from CMB ($\Delta N_{\rm eff}<0.2$) [63].
The fact that $\Delta N_{\rm eff}\neq 0$ in both scenarios fits with the non-
flatness results indicated by $\Omega_{k0}$. In fact, a closed Universe
$\Omega_{k0}$ has been found in several different analyses of Hořava-Lifshitz
cosmology [42, 44, 43], and it now seems that the model does indeed prefer a
closed Universe. Other studies have reported a strong preference for a closed
Universe in the Planck data [64], a result which is sensitive to the amount of
lensing in the sample. Other data sets, primarily BAO, strongly favour a
closed Universe (under the assumption of $\Lambda$CDM) to the extent that the
tension in $\Omega_{k0}$ has been estimated at $2.5-3\sigma$ [47].

##  IV Discussion & Conclusions

The focus of this paper is exploring the significance of the parameter
$\lambda$ in HL cosmology, which is believed to play a crucial role in
determining or characterizing the extent of Lorentz violation present in the
theory. Our primary objective was to examine the impact of incorporating or
omitting data sources corresponding to different energy levels on the
estimated value of $\lambda$. In other words, we wanted to investigate whether
there is a correlation between the energy level and the value of $\lambda$.

In the detailed-balance scenario, our findings indicate that the parameter
$\lambda$ consistently exceeds unity at a 1$\sigma$ level; however, if we
exclude low-energy sources, such as Hubble parameter measurements, the value
of $\lambda$ tends to shift closer to the ultraviolet region. In the case of
beyond detailed balance, we observed a mean value below unity in two
instances. While the relationship between energy flow and $\lambda$ is not as
evident here as in the detailed-balance scenario, we can still observe a clear
trend: when the lowest-energy data is removed, $\lambda$ is pushed below
unity. Moreover, we have performed the same calculations when considering the
hard prior $\lambda\geq 1$, since analytical considerations like [5, 48]
demonstrate that the parameter range $1>\lambda>1/3$ is very problematic and
results in ghost instabilities of an also problematic scalar mode. For that
reason that range is therefore considered unphysical, however there is not yet
an efficient mechanism proposed for getting rid of that problematic mode in
the GR limit, which due to its higher symmetry contains only tensor modes.
There are also strong observational bounds on the existence of the HL scalar
mode [65, 66, 25]; therefore, potential quantum instabilities of a non-
observed mode and no mechanism available for suppressing it in the IR requires
careful consideration before drawing any conclusions. That is why we present
general results both with and without that prior on $\lambda$ in order to have
a more complete picture of the theory. In the case when the prior $\lambda\geq
1$ is imposed, excluding low energy data changes the BDB case more, since this
model has a stronger tendency to flow to $\lambda<1$ without a prior, but also
the DB case sees changes, especially on the form of larger errors on the case
with all data, although the reason for this is not clear.

The phenomenological results obtained suggest the potential existence of
Lorentz violation in astrophysical probes at higher energies; however, it is
important to note that the existing bounds on Lorentz violation in the matter
and electromagnetic sectors are significantly stronger than those in the
gravity sector [67, 68, 69]. Several theoretical studies have been dedicated
to addressing this issue [70, 71, 72, 73], proposing additional terms to the
action aimed at preventing the leakage of symmetry violations from the gravity
sector to the matter sector or suppressing them in the low-energy regime of
the matter sector. Moreover, there are theoretical perspectives [74, 75, 76]
which question whether non-unity values of $\lambda$ necessarily indicate
departures from classical general relativity. Therefore, it is crucial to
exercise caution when interpreting the theoretical implications of the results
presented above. A natural question which also arises is how the parameter
$\lambda$ would vary in different non-projectable extensions of HL gravity,
which have been proposed to address some of its shortcomings. While work on
these scenarios is ongoing, it is important to note that relaxing
projectability conditions leads to a more complex theory with a higher number
of parameters and interpretation issues. For now, we present our current
findings, which are intriguing and merit further theoretical and observational
investigation. These results could offer new insights into the underlying HL
theory.

###### Acknowledgements.

NAN is thankful for financial support by CNES, and for discussions with
Vincenzo Salzano.

##  Appendix A The data

###  A.1 Planck CMB

The non-perturbative CMB information is contained in the shift parameters,
describing the location of the first peak in the temperature angular power
spectrum. Here, we use the shift parameters extracted from the final Planck
2018 data release [59]. For a vector $\bm{\theta}$ containing the model
parameters, the geometrical CMB shift parameters read

$R(\bm{\theta})=100\sqrt{\Omega_{m0}h^{2}}\frac{d_{A}^{c}(z_{*},\bm{\theta})}{c},\quad\ell_{a}(\bm{\theta})=\pi\frac{d_{A}^{c}(z_{*},\bm{\theta})}{r_{s}(z_{*},\bm{\theta})},$
(12)

which together with $\omega_{b}=\Omega_{b0}h^{2}$ makes up the CMB distance
priors. In the above equations, $d_{A}^{c}(z_{*},\bm{\theta})$ and
$r_{s}(z_{*},\bm{\theta})$ are the comoving angular-diameter distance and the
sound horizon, respectively, defined as

$d_{A}^{c}(z,\bm{\theta})=\begin{cases}\frac{c}{H_{0}}\frac{1}{\sqrt{\Omega_{k0}}}\sinh{\left[\sqrt{\Omega_{k0}}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime},\bm{\theta})}\right]},&\Omega_{k0}>0\\\
\frac{c}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime},\bm{\theta})},&\Omega_{k0}=0\\\
\frac{c}{H_{0}}\frac{1}{\sqrt{|\Omega_{k0}|}}\sinh{\left[\sqrt{|\Omega_{k0}|}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime},\bm{\theta})}\right]},&\Omega_{k0}>0,\end{cases}$
(13)

and

$r_{s}(z,\bm{\theta})=H_{0}\int_{z}^{\infty}\frac{c_{s}(z^{\prime})dz^{\prime}}{E(z^{\prime},\bm{\theta})},$
(14)

where $E(z,\bm{\theta})\equiv H(z,\bm{\theta})/H_{0}$ and $c_{s}(z)$ is the
sound speed

$c_{s}(z)=\frac{c}{\sqrt{3[1+R_{b}/(1+z)]}},\quad
R_{b}=31500\omega_{b}\left(\frac{T_{\rm CMB}}{2.72}\right)^{-4}.$ (15)

The above relations are defined at the photon-decoupling redshift $z_{*}$,
which is defined as [77]

$\displaystyle z_{*}=$ $\displaystyle
1048\left[1+0.00124\omega_{b}^{-0.738}\right]\left[1+g_{1}\omega_{b}^{g_{2}}\right]$
(16) $\displaystyle g_{1}=$ $\displaystyle
0.0783\omega_{b}^{-0.238}\left[1+39.5\omega_{b}^{-0.763}\right]^{-1}$
$\displaystyle g_{2}=$ $\displaystyle
0.560\left[1+21.1\omega_{b}^{1.81}\right]^{-1}.$

In the model we are considering, the number of relativistic species are no
longer equal to the $\Lambda$CDM value of $N_{\rm eff}=3.046$, and since the
value of this quantity have several effects on the CMB, we need now to
incorporate this into the analysis. Following the procedure in [78], we add
the CDM density parameter $\omega_{c}=(\Omega_{m0}-\Omega_{b0})h^{2}$ and
$N_{\rm eff}=3.046+\Delta N_{\rm eff}$, and the full CMB distance priors read
$\bm{v}=(R,\ell_{a},\omega_{b},\omega_{c},N_{\rm eff})$. We use the values
found in [78]

$\bm{v}=\begin{pmatrix}1.7661\\\ 301.7293\\\ 0.02191\\\ 0.1194\\\
2.8979,\end{pmatrix}$ (17)

as well as the associated covariance matrix

$C_{\bm{v}}=10^{-8}\begin{pmatrix}33483.54&-44417.15&-515.03&-360.42&-274151.72\\\
-44417.15&4245661.67&2319.46&63326.47&4287810.44\\\
-515.03&2319.46&12.92&51.98&7273.04\\\
-360.42&63326.47&51.98&1516.28&92013.95\\\
-274151.72&4287810.44&7273.04&92013.95&7876074.60\end{pmatrix}$ (18)

and the CMB $\chi^{2}$ finally reads

$\chi^{2}_{\rm CMB}=\left(\Delta\bm{v}\right)^{\rm
T}C_{\bm{v}}^{-1}\Delta\bm{v},$ (19)

where $\Delta\bm{v}$ is the difference between the theoretical and observed
values of the distance priors $\bm{v}$.

###  A.2 Pantheon+ and Cepheid variable stars

The Pantheon+ sample is a set of 1701 light curves of 1550 distinct Supernovae
Type Ia (SNeIa) in the redshift range $0.001<z<2.26$ [79, 80]. Compared to the
previous catalogue Pantheon, this update features an increased redshift range
of cross-calibrated photometric systems of sources and improved treatment of
systematic effects; all together, this results in an factor of 2 improvement
in cosmological constraining power [80]. Also contained in this catalogue are
those SNeIa from host galaxies with known Cepheid distances, providing a
robust calibration of the SNeIa light curve (known as “anchoring”) and enables
the simultaneous determination of expansion-history parameters (for example
$\Omega_{m0}$) and the local expansion rate $H_{0}$, which are degenerate when
using SNeIa alone.

In order to form the $\chi^{2}$ for Pantheon, we write down the luminosity
distance as (all details can be found in [80])

$d_{L}(z,\bm{\theta})=(1+z_{\rm hel})\,d_{A}^{c}(z,\bm{\theta}),$ (20)

where $z_{\rm hel}$ is the redshift measured in the heliocentric frame. From
this we can define distance modulus as

$\mu(z,\bm{\theta})=m(z,\bm{\theta})-M,$ (21)

where $m$ and $M$ is the apparent and fiducial absolute magnitude,
respectively. From this expression, we can write the apparent magnitude as

$m(z,\bm{\theta})=5\log{d_{L}(z,\bm{\theta})}+25+M,$ (22)

where $d_{L}(z,\bm{\theta})$ is now expressed in Mpc. Note here that we are
not able to simply write the distance modulus as
$\mu=m-M=5\log{d_{L}}+\mu_{0}$ and marginalize over $\mu_{0}$; when using
SNeIa alone, $M$ can indeed be marginalized over, but including Cepheids
breaks the degeneracy between $M$ and $H_{0}$, and the fiducial absolute
magnitude $M$ becomes an extra free parameter in our numerical analysis. As
such, we write the distance residuals as [80]

$\Delta D_{i}=\begin{cases}\mu_{i}-\mu_{i}^{C},&i\in\text{Cepheids}\\\
\mu_{i}-\mu_{i}^{\rm model}&\text{others},\end{cases}$ (23)

where the theoretical value is replaced by the corresponding Cepheid-
calibrated value for the host galaxy distance, hence providing the anchoring
and breaking the degeneracy between $\Omega_{m0}$ and $M$. The statistical and
systematic uncertainties are contained in the covariance matrix $C^{\rm
SN+Cepheids}_{\rm stat+sys}$, and the $\chi^{2}$ measure becomes

$\chi^{2}_{\rm Pantheon+Cepheids}=(\Delta\bm{D})^{\rm T}\left(C^{\rm
SN+Cepheids}_{\rm stat+sys}\right)^{-1}(\Delta\bm{D})$ (24)

###  A.3 Gamma-ray bursts

Gamma-ray bursts are one of the most energetic events in the Universe, and may
be used as a probe complementing SNeIa [81]. This sample consists of 79 long
gamma-ray bursts (the Mayflower sample) calibrated using the Padé approximant
method. This approach avoids the circularity problem usually present when
trying to use gamma-ray bursts as cosmological rulers [82]. The $\chi^{2}$ for
this dataset is found by considering the distance modulus (as for SNeIa, but
we can now marginalize over $\mu_{0}$), and we write the $\chi^{2}$ as

$\chi^{2}_{\rm GRB}=a+\log{\frac{e}{2\pi}}-\frac{b^{2}}{e},$ (25)

where $a=(\Delta\mu)^{\rm T}C^{-1}_{\rm GRB}(\Delta\mu)$, $b=(\Delta\mu)^{\rm
T}C^{-1}_{\rm GRB}\cdot\bm{1}$, and $c=\bm{1}^{\rm T}\cdot C^{-1}_{\rm
GRB}\cdot\bm{1}$. Here we have defined $\Delta\mu=\mu_{\rm theory}-\mu_{\rm
obs}$ where $\mu_{\rm obs}$ is the observed distance modulus, and $\mu_{\rm
theory}$ comes from the Padé method after calibration. The full details of
this calibration and the data can be found in [81].

###  A.4 Cosmic Chronometers (CC)

We use Hubble-parameter measurements from Passively-evolving Early-Type
Galaxies (ETG), which have low star-formation rate and old stellar
populations. The spectral properties of ETGs can be traced along cosmic time
$t$ by measuring the Hubble parameter $H(z)=dz/dt(1+z)$ independent of the
cosmological model [83], making them a type of standardisable clock, or cosmic
chronometer. We use a sample covering the redshift range $0<z<1.97$ [84, 85,
86]. To construct a covariance matrix for the CC data points we follow the
procedure in [87, 88], which includes the following sources of uncertainty

$C_{\rm CC}=C_{\rm CC}^{\rm stat}+C_{\rm CC}^{\rm young}+C_{\rm CC}^{\rm
model}+C_{\rm CC}^{\rm met},$ (26)

where $C_{\rm CC}^{\rm stat}$, $C_{\rm CC}^{\rm young}$, and $C_{\rm CC}^{\rm
model}$ correspond to uncertainty from statistical errors, sample-
contamination by younger (and hotter) stars, model dependence, and stellar
metallicity. In these considerations, the largest source of error comes from
the $C_{\rm CC}^{\rm young}$ contribution [88]. The model uncertainty $C_{\rm
CC}^{\rm model}$ can be further broken down into

$C_{\rm CC}^{\rm model}=C_{\rm CC}^{\rm SFH}+C_{\rm CC}^{\rm IMF}+C_{\rm
CC}^{\rm st.lib.}+C_{\rm CC}^{\rm SPS},$ (27)

denoting star-formation history (SFH), initial mass function (IMF), stellar
library (st.lib.), and stellar population synthesis (SPS). We use the
accompanying code555https://gitlab.com/mmoresco/CCcovariance [88] to generate
the final covariance matrix for our data points, after which the $\chi^{2}$
reads

$\chi^{2}_{\rm CC}=(\Delta H)^{\rm T}\,C_{\rm CC}^{-1}\,(\Delta H),$ (28)

where $(\Delta H)_{i}=H_{\rm theory}(z_{i})-H_{\rm obs}(z_{i})$.

###  A.5 Baryon Acoustic Oscillations

We include seven different BAO data sets in our analysis, as shown below

WiggleZ:  We include data from the WiggleZ Dark Energy Survey at redshift
points $z_{W}=\\{0.44,0.6,0.73\\}$ [89]. Here, the observables are

$A(x,\bm{\theta})=100\sqrt{\omega_{m}}\frac{D_{V}(z,\bm{\theta})}{cz},\quad
F(z,\bm{\theta})=\frac{d_{A}^{c}(z,\bm{\theta})H(z,\bm{\theta})}{c},$ (29)

where $A(x,\bm{\theta})$ is the acoustic parameter, $F(z,\bm{\theta})$ is the
Alcock-Paczynski distortion parameter, and $\omega_{m}=\Omega_{m0}h^{2}$.
$D_{V}(z,\bm{\theta})$ is the volume distance, defined as

$D_{V}(z,\bm{\theta})=\left[d_{A}^{c}(z,\bm{\theta})^{2}\frac{cz}{H(z,\bm{\theta})}\right]^{1/3},$
(30)

and we find the $\chi^{2}$ as

$\chi^{2}_{W}=(\Delta\mathcal{F}_{W})^{\rm T}C^{-1}_{W}\Delta\mathcal{F}_{W},$
(31)

where $\Delta\mathcal{F}_{W}=\mathcal{F}_{W,\rm theory}-\mathcal{F}_{W,\rm
obs}$, and $\mathcal{F}_{W}=\\{A(z_{W}),F(z_{W})\\}$. For all the other BAO
probes, we find the $\chi^{2}$ in the same way.

SDSS-BOSS:  we include data from the SDSS-II BOSS DR12 and SDSS-IV DR16 LRG
(Luminous Red Galaxy growth-rate sample) at redshifts $z_{\rm
B}=\\{0.38,0.51,0.61\\}$. For this data, we use the quantities

$d_{A}^{c}(z,\bm{\theta})\frac{r_{s}^{\rm
fid}(z_{d})}{r_{s}(z_{d},\bm{\theta})},\quad
H(z,\bm{\theta})\frac{r_{s}(z_{d},\bm{\theta})}{r_{s}^{\rm fid}(z_{d})},$ (32)

evaluated at the dragging redshift $z_{d}$, which we approximate as [90]

$\displaystyle z_{d}=$ $\displaystyle
1291\frac{\omega_{m}^{~{}0.251}}{1+0.659\omega_{m}^{~{}0.828}}\left(1+b_{1}\omega_{b}^{~{}b_{2}}\right)$
(33) $\displaystyle b_{1}=$ $\displaystyle
0.313\omega_{m}^{~{}-0.419}\left(1+0.607\omega_{m}^{~{}0.6748}\right)$
$\displaystyle b_{2}=$ $\displaystyle 0.238\omega_{m}^{~{}0.223}.$

In Eq.(32) $r_{s}^{\rm fid}$ is the sound horizon at the dragging redshift
evaluated for a fiducial cosmological model; here, we take $r_{s}^{\rm
fid}(z_{d})=147.78$ Mpc [91, 92].

SDSS QSO:  From the SDSS IV BOSS DR14 quasar sample, we have the following
data points at redshifts $z_{Q}=\\{0.978,1.23,1.526,1.944\\}$ [93]

$\displaystyle d_{A}^{c}\frac{r_{s}^{\rm fid}(z_{d})}{r_{s}(z_{d}}=$
$\displaystyle\\{1586.18\pm 284.93,1769.08\pm 159.67,1768.77\pm
96.59,1807.98\pm 146.46\\}$ (34) $\displaystyle
H\frac{r_{s}(z_{d})}{r_{s}^{\rm fid}(z_{d})}=$ $\displaystyle\\{113.72\pm
14.63,131.44\pm 12.42,148.11\pm 12.75,172.63\pm 14.79\\}$ $\displaystyle
D_{V}\frac{r_{s}^{\rm fid}(z_{d})}{r_{s}(z_{d})}=$ $\displaystyle\\{2933.59\pm
327.71,3522.04\pm 192.74,3954.31\pm 141.71,4575.17\pm 241.61\\}.$

From the eBOSS DR16 QSO release [94, 95], we also have the following data
points at redshift $z=1.480$

$\frac{c}{Hr_{s}(z_{d})}=13.23\pm
0.47,\quad\frac{d_{A}^{c}}{r_{s}(z_{d})}=30.21\pm 0.79.$ (35)

eBOSS ELG:  from the eBOSS DR16 Emission Line Galaxy sample (ELG) we have, at
the effective redshift $z_{\rm eff}=0.845$ [96, 97]

$\frac{c}{Hr_{s}(z_{\rm eff})}=19.6^{+2.2}_{-2.1},\quad\frac{d_{A}^{c}(z_{\rm
eff})}{r_{s}(z_{d})}=19.5\pm 1$ (36)

eBOSS CMASS:  from void-galaxy cross-correlations in redshift-space distortion
corrected data from the DR12 LRG CMASS sample, we have, at the effective
sample redshift of $z_{\rm eff}=0.69$ the following [98]

$\frac{c}{Hr_{s}(z_{\rm eff})}=17.48\pm 0.23,\quad\frac{d_{A}^{c}(z_{\rm
eff})}{r_{s}(z_{d})}=20.10\pm 0.34.$ (37)

Lyman-$\alpha$:  using the autocorrelation of Lyman-$\alpha$ absorption in
quasars and Lyman-$\alpha$ cross correlation in the eBOSS DR16 quasar sample
[99] we have (at the effective redshift $z_{\rm eff}=2.33$)

$\frac{c}{Hr_{s}(z_{\rm eff})}=8.99\pm 0.19,\quad\frac{d_{A}^{c}(z_{\rm
eff})}{r_{s}(z_{d})}=37.5\pm 1.1.$ (38)

All the $\chi^{2}$ measures for the above BAO points are included in our
analysis.

##  Appendix B All parameter constraints

| Detailed balance, prior $\lambda\geq 1$ | Detailed balance, no prior on $\lambda$
---|---|---
Parameter |  CMB+BAO+H(z) +SNe1a+Cepheids +GRB |  CMB+BAO +SNe1a+Cepheids +GRB |  CMB+BAO |  CMB+BAO+H(z) +SNe1a+Cepheids +GRB |  CMB+BAO +SNe1a+Cepheids +GRB |  CMB+BAO
$\Omega_{b}$ | $0.0049768\pm 0.0000016$ | $0.04898^{+0.00055}_{-0.000053}$ | $0.05104\pm 0.00080$ | $0.049236\pm 0.000070$ | $0.04907\pm 0.00058$ | $0.05139^{+0.00019}_{-0.00020}$
$\Omega_{b}h^{2}$ | $0.020952\pm 0.000074$ | $0.02227\pm 0.00018$ | $0.02192^{+0.00019}_{-0.00021}$ | $0.02158\pm 0.00012$ | $0.02217\pm 0.00017$ | $0.02187\pm 0.00011$
$\Omega_{m}$ | $0.3336\pm 0.0018$ | $0.3153^{+0.0057}_{-0.0055}$ | $0.3398\pm 0.0083$ | $0.3204\pm 0.0030$ | $0.3170\pm 0.0056$ | $0.34437^{+0.00010}_{-0.00012}$
$\Omega_{m}h^{2}$ | $0.14043^{+0.00024}_{-0.00025}$ | $0.14335\pm 0.00090$ | $0.1458^{+0.0011}_{-0.0010}$ | $0.14047^{+0.00030}_{-0.00033}$ | $0.14323\pm 0.00092$ | $0.14657\pm 0.00023$
$\Omega_{k}10^{4}$ | $-4.1364\pm 0.0040$ | $-6.116^{+1.55}_{-0.30}$ | $-11.60^{+2.61}_{-1.76}$ | $-4.254\pm 0.019$ | $-5.745\pm 0.029$ | $-13.5030^{+0.0088}_{-0.0086}$
$\Omega_{r}10^{5}$ | $9.937^{+0.036}_{-0.035}$ | $9.20\pm 0.12$ | $9.74^{+0.19}_{-0.18}$ | $9.543^{+0.070}_{-0.065}$ | $9.259^{+0.012}_{-0.011}$ | $9.83^{+0.012}_{-0.018}$
$h$ | $0.6488\pm 0.0012$ | $0.6743^{+0.0043}_{-0.0044}$ | $0.6553^{+0.0061}_{-0.0063}$ | $0.6621^{+0.0023}_{-0.0024}$ | $0.6722\pm 0.0040$ | $0.65239\pm 0.00040$
$M$ | $-19.5051\pm 0.0013$ | $-19.437^{+0.012}_{-0.013}$ | - | $-19.4783\pm 0.0075$ | $-19.442\pm 0.012$ | -
$\Delta N_{\rm eff}$ | $0.0046750\pm 0.0000076$ | $\bm{0.1104^{+0.0011}_{-0.0049}}$ | $0.038^{+0.013}_{-0.015}$ | $0.005099\pm 0.000060$ | $0.009670\pm 0.000062$ | $0.05195\pm 0.00019$
$\lambda$ | $1.0406\pm 0.0023$ | $\bm{<1.0032}^{\dagger}$ | $1.0146^{+0.055}_{-0.053}$ | $1.02726\pm 0.00012$ | $1.0065\pm 0.0018$ | $1.0159\pm 0.0014$
$\left|\tfrac{G_{\rm cosmo}}{G_{\rm grav}}-1\right|$ | $0.0574\pm 0.0030$ | $\bm{<0.0035}^{\dagger}$ | $0.0214^{+0.0078}_{-0.0077}$ | $0.03928\pm 0.00017$ | $0.00997^{+0.0026}_{-0.0025}$ | $0.0232\pm 0.0020$
$\chi^{2}_{\rm{min}}$ | $1778.27$ | $1635.41$ | $27.30$ | $1705.04$ | $1638.49$ | $27.76$
Table 1: Parameter constraints at $1\sigma$ for the Detailed Balance case, with and without a hard prior on the parameter $\lambda$. $\dagger$ implies a one-sided upper bound resulting from a hard uniform prior, and bold indicates a particularly noisy parameter.  | Beyond detailed balance, prior $\lambda\geq 1$ | Beyond detailed balance, no prior on $\lambda$
---|---|---
Parameter |  CMB+BAO+H(z) +SNe1a+Cepheids +GRB |  CMB+BAO +SNe1a+Cepheids +GRB |  CMB+BAO |  CMB+BAO+H(z) +SNe1a+Cepheids +GRB |  CMB+BAO +SNe1a+Cepheids +GRB |  CMB+BAO
$\Omega_{b}$ | $0.049371^{+0.00049}_{-0.00048}$ | $0.05024\pm 0.00049$ | $0.05116^{+0.00062}_{-0.00063}$ | $0.04900\pm 0.00040$ | $0.05034^{+0.00017}_{-0.00016}$ | $0.05034\pm 0.00016$
$\Omega_{b}h^{2}$ | $0.022922\pm 0.00023$ | $0.02262\pm 0.00020$ | $0.02226\pm 0.00018$ | $0.022864\pm 0.00011$ | $0.02250\pm 0.00011$ | $0.02250\pm 0.00011$
$\Omega_{m}$ | $0.3198\pm 0.0053$ | $0.3194^{+0.0056}_{-0.0054}$ | $0.3349\pm 0.0073$ | $0.3109^{+0.0038}_{-0.0041}$ | $0.3232\pm 0.0031$ | $0.3284^{+0.0093}_{-0.0091}$
$\Omega_{m}h^{2}$ | $0.1484\pm 0.0017$ | $0.14386^{+0.00095}_{-0.00096}$ | $0.1458\pm 0.0011$ | $0.14509^{+0.00015}_{-0.00016}$ | $0.14442^{+0.00046}_{-0.00049}$ | $0.1458^{+0.0030}_{-0.0027}$
$\Omega_{k}10^{3}$ | $-9.71^{+1.73}_{-1.83}$ | $-4.98^{+0.77}_{-0.46}$ | $-3.93\pm 0.15$ | $-5.399^{+0.022}_{-0.023}$ | $-4.338\pm 0.080$ | $-6.32\pm 2.56$
$\Omega_{r}10^{5}$ | $9.01\pm 0.13$ | $9.29\pm 0.11$ | $9.61^{+1.63}_{-1.61}$ | $8.97^{+1.06}_{-1.11}$ | $9.36\pm 0.61$ | $9.431^{+0.028}_{-0.030}$
$h$ | $0.6813\pm 0.0048$ | $0.6711^{+0.0039}_{-0.0040}$ | $0.6599^{+0.0056}_{-0.0055}$ | $0.6831^{+0.0043}_{-0.0040}$ | $0.6685^{+0.0022}_{-0.0021}$ | $0.6660^{+0.0109}_{-0.0096}$
$M$ | $-19.414^{+0.014}_{-0.015}$ | $-19.446\pm 0.0011$ | - | $-19.412^{+0.012}_{-0.011}$ | $-19.4525\pm 0.0040$ | -
$\Delta N_{\rm eff}$ | $0.61\pm 0.14$ | $0.258^{+0.028}_{-0.048}$ | $0.198\pm 0.015$ | $0.2578^{+0.0042}_{-0.0044}$ | $0.2166\pm 0.0070$ | $0.31^{+0.19}_{-0.15}$
$\lambda$ | $1.00644\pm 0.00020$ | $<1.0068^{\dagger}$ | $1.0065\pm 0.0025$ | $0.9949^{+0.0045}_{-0.0046}$ | $1.00578\pm 0.00086$ | $0.9972^{+0.0081}_{-0.0088}$
$\left|\tfrac{G_{\rm cosmo}}{G_{\rm grav}}-1\right|$ | $0.00957\pm 0.00029$ | $<0.010^{\dagger}$ | $0.0096\pm 0.0037$ | $0.0078^{+0.0070}_{-0.0053}$ | $0.0086\pm 0.0013$ | $<0.019$
$\omega_{1}$ | $0.69957\pm 0.00039$ | $0.6909^{+0.0063}_{-0.0062}$ | $0.6789^{+0.0070}_{-0.0075}$ | $0.6866\pm 0.0036$ | $0.6898^{+0.0044}_{-0.0043}$ | $0.6898\pm 0.0043$
$\omega_{3}10^{6}$ | $7.07^{+1.62}_{-1.71}$ | $1.77^{+0.10}_{-0.15}$ | $1.5201^{+0.068}_{-0.071}$ | $2.9109^{+0.047}_{-0.0049}$ | $1.80\pm 0.15$ | $1.79\pm 0.15$
$\chi^{2}_{\rm{min}}$ | $1634.37$ | $1632.36$ | $23.82$ | $1635.54$ | $1633.85$ | $21.85$
Table 2: Parameter constraints at $1\sigma$ for the Beyond Detailed Balance
case, with and without a hard prior on the parameter $\lambda$. $\dagger$
implies a one-sided upper bound resulting from a hard uniform prior.

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Comment on “Semiconducting Layered Blue Phosphorus: A Computational Study”

Using results drawn from the pages of PRLZhu2014 , the authors comment on
“Semiconducting Layered Blue Phosphorus: A Computational Study”. In recent
letterZhu2014  unknown phase of phosphorus with high stability and a wide
fundamental gap was proposed by Zhu, and Tománek. The first half of the
comment questioned elements of phosphorus blue structure, while the second
half devoted to van der Waals forces between AB stacked blue phosphorus layers
which missed by them.

Converting a mono-layer of black to blue phosphorus, they used dislocation
with constantlocal band angles and changed z lattice vector direction. This
dislocation is not complete method for predicting new structure, While
predicting a new system is available via ab initio evolutionary algorithm
implemented for instance in USPEX code Uspex1 ; Uspex2 ; Uspex3 .

Using USPEX code accompanied with VASP Kresse2007 , We found black phosphorous
as the most stable structure and after that the blue phosphorous is stable
with the difference energy of 2.62 eV. The interlayer distance, dint, in above
structure from USPEX prediction for Black and Blue ph. is 5.78 and 3.56 Å that
is near to our results from ven der Waals calculation in table 1. Then in
monolayer structure, we found black monolayar is 0.034 eV stable than blue
phosphorous monolayer.

Figure 1:  Total interaction energy between two layer in AB and AA stacked
blue phosphorous separated by the zero Å thick vacuum.

Presented here, reveals, the interlayer distance adopted in the letter are not
in fact the most stable distance. The phosphorus layers, especially in blue
configuration, stacked together by van der Waals interactions, like
graphiteLikai2014 .

The Brillouin zone of primitive cell for all calculation was sampled using
10$\times$10$\times$1 k points in standard DFT calculation pseudo-potential
methods by Quantum Espresso (QE) (ecutwfc=40.0 and ecutrho=500) QE-2009 .

Table 1: The nearest distance between the adjacent layer (dint) interlayer distance for a, zero vacuum, and b, 15 Å vacuum,. $\Delta$E${}^{{}^{\prime}}$(Ry*10-2) is, $\Delta$E, energy difference between minimum energy level and straight energy level per atom. Values in parenthesis is related to black phosphorus. name | a | a | b | b
---|---|---|---|---
| dint(Å) |  $\Delta$E${}^{{}^{\prime}}$ | dint(Å) |  $\Delta$E${}^{{}^{\prime}}$
AB-PBE | 3.26(5.40) | 1.35(0.20) | 3.06(5.82) | 0.99(0.02)
AB-vdW-DF | 3.46(5.62) | 1.30(0.46) | 3.12(5.86) | 0.69(0.17)
AA-PBE | 3.44(5.24) | 1.26(0.39) | 4.12(5.86) | 0.10(0.03)
AA-vdW-DF | 3.60(5.46) | 1.40(0.59) | 4.24(5.88) | 0.32(0.18)

Concerning van der Waals (vdW) interaction via self consistent vdW-DF
implemented in QE code, We found 3.46 and 3.12 Å as interlayer distance for
two periodic array of AB stacked slab separated by a viz. a) zero and b)15 Å
thick vacuum region, respectively. This distance is completely different of
5.63 angstrom mentioned in FIG. 1d of Letter Zhu2014 . In same calculation for
AA stacked, dint is 3.60 and 4.24 Å (see table 1 and figure 1). we repeated
this calculation for black phosphorus, so interlayer distance reached its
minimum in $\simeq$ 5.6 Å, in accordance with experimental results Likai2014 .
The difference between minimum energy level and straight energy level in black
one is around 0.5*10-2 Ry, while in blue one is 1.30*10-2 Ry.

I. Abdolhosseini Sarsari1, Z. Allahyari1,2,3, M. Alaei1 and S. Sanvito4
1Department of Physics, Isfahan University of Technology, Isfahan,
84156-83111, Iran
2Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region,
141700, Russia
3Skolkovo Institute of Science and Technology, Skolkovo Innovation Center,
Bldg. 3, Moscow 143026, Russia
4School of Physics, AMBER and CRANN, Trinity College, Dublin 2, Ireland

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