# MuTox: Universal MUltilingual Audio-based TOXicity
Dataset and Zero-shot Detector

Marta R. Costa-jussà, Mariano Coria Meglioli, Pierre Andrews,
David Dale, Prangthip Hansanti, Elahe Kalbassi,
Alex Mourachko, Christophe Ropers, Carleigh Wood
FAIR, Meta
{costajussa,mfcoria,mortimer,daviddale,prangthiphansanti,
<EMAIL_ADDRESS>
###### Abstract

Research in toxicity detection in natural language processing for the speech
modality (audio-based) is quite limited, particularly for languages other than
English. To address these limitations and lay the groundwork for truly
multilingual audio-based toxicity detection, we introduce MuTox, the first
highly multilingual audio-based dataset with toxicity labels. The dataset
comprises 20,000 audio utterances for English and Spanish, and 4,000 for the
other 19 languages. To demonstrate the quality of this dataset, we trained the
MuTox audio-based toxicity classifier, which enables zero-shot toxicity
detection across a wide range of languages. This classifier outperforms
existing text-based trainable classifiers by more than 1% AUC, while expanding
the language coverage more than tenfold. When compared to a wordlist-based
classifier that covers a similar number of languages, MuTox improves precision
and recall by approximately 2.5 times. This significant improvement
underscores the potential of MuTox in advancing the field of audio-based
toxicity detection.

Warning: This article includes examples of language that can be considered
offensive or upsetting.

##  1 Introduction

Text toxicity detection has been largely explored for different tasks in
Natural Language Processing (NLP) Kaggle (2018). Wordlist-based toxicity
classifiers—e.g., etox Costa-jussà et al. (2023)— scale well to a large number
of languages NLLB Team et al. (2022) and context-based classifiers are able to
detect beyond lexical toxicity with tools such as detoxify
111https://github.com/unitaryai/detoxify.

When exploring audio-based toxicity detection, there are either cascaded
systems which extend text toxicity detection with speech recognition Seamless
Communication et al. (2023a); or end-to-end audio-based toxicity
classification Ghosh et al. (2021) which provides an English dataset together
with end-to-end toxicity detection results. This work shows that gains of
English text-less audio-based classifiers over text-based classifiers are
specially relevant when applied to out-of-domain, coherently with the previous
study on a non-disclosed dataset Yousefi and Emmanouilidou (2021).

In this paper, we go far beyond existing research in audio-based toxicity
detection by providing the first highly multilingual audio-based toxicity
annotated dataset (MuTox dataset, 21 languages) together with the first text-
less massive multilingual metric (MuTox classifier, 100+ languages). Note that
multilinguality for audio-based toxicity detection becomes even more crucial
for the task of added toxicity in the context of multimodal and multilingual
translation, where the case of adding or deleting toxicity may be considered
as a critical error Seamless Communication et al. (2023a).

In particular, the main contributions of this paper are: providing guidelines
for audio-based toxicity annotation (section 3); releasing the first highly
multilingual audio-based toxicity dataset and benchmark with human annotations
for 21 languages (section 4); analyzing the performance of text-based
classifiers when applied to audio-based toxicity detection (section 6);
proposing MuTox, a massively multilingual audio-based toxicity classifier
(section 5).

Our results can be summarized as follows:

  * •

When compared to the strongest performing systems, which are composed of
speech recognition plus trainable text-based toxicity detectors, MuTox
slightly surpasses (+1% on average), while offering more than 10 times the
language coverage.

  * •

When compared to systems with the highest coverage, which are composed of
speech recognition plus wordlists toxicity detectors, MuTox improves precision
and recall by a factor of approximately 2.5.

These results highlight the effectiveness of MuTox in multilingual audio-based
toxicity detection, demonstrating its potential to significantly advance the
field.

##  2 Background

#### Text-based toxicity classifiers.

In this work, we use etox Costa-jussà et al. (2023) and detoxify as primary
text-based toxicity detectors. etox is chosen for offering the widest language
coverage in toxicity detection. detoxify is chosen for being one of the
available tools with the highest performance on several JigSaw benchmarks with
a single model Thewlis et al. (2021).

etox is an open-
source222https://github.com/facebookresearch/seamless_communnication/toxicity
wordlist-based classifier covering 200 languages. While this metric has
several advantages - namely, it covers a massive number of languages and is
highly transparent - it has other limitations such as only detecting lexical
toxicity and not distinguishing polysemous words which may be toxic in some
contexts and not others.

detoxify is a text-based toxicity classifier trained mainly in JigSaw Kaggle
(2018) which is freely available in 8 languages (English, French, German,
Italian, Portuguese, Russian, Spanish, Turkish).

#### Datasets

A well-known resource for textual toxicity detection is the JigSaw dataset
Kaggle (2018), which consists of a large number of Wikipedia comments which
have been labeled by human raters for toxic meaning. This dataset has been
used for several Kaggle competitions covering a broad range of tasks, from
detecting types of toxicity to analyzing bias in toxicity detection systems.
There are also other related datasets e.g. Röttger et al. (2022).

We are not aware of any specific multilingual speech datasets which contain
toxicity or toxicity labels. However, we want to pre-select data for
annotation to maximize chances of the annotators confirming toxicity.
Relatedly, Detoxy Ghosh et al. (2021) estimated the amount of toxicity for
several English spoken datasets using text-based classifiers. In this work, we
follow a similar approach, by exploring two highly multilingual spoken
datasets: CommonVoice Ardila et al. (2020) and SeamlessAlign Seamless
Communication et al. (2023b).

CommonVoice is a massively-multilingual collection of transcribed speech
intended for speech technology research and development. SeamlessAlign is an
automatically collected pairs of natural speech from raw web corpora through
parallel data mining following methodology described in Seamless Communication
et al. (2023a). Additionally, for English and Spanish, we use
SeamlessAlignExpressive which extends the process used for SeamlessAlign to
create a large collection of multilingual speech/speech and speech/text pairs,
aligned not only in meaning but also expressivity. Text and audio sources are
identical to SeamlessAlign. The modified mining algorithm is described in
section 4 from Seamless Communication et al. (2023b).

#### Experimental Task.

Toxicity detection in natural language processing is the task of assigning a
toxicity label to a speech or text utterance.

##  3 Annotation Guidelines

This section reports the detailed guidelines that we provide to annotators to
detect toxic content in audio speech at the level of single utterances. This
toxicity could be due to aspects of lexical semantics or of perlocutionary
effects. For this we provide annotators with the following definitions:

  * •

Utterance refers to a unit of audio speech that is comparable to what a
sentence is for writing.

  * •

Lexical semantics refers to meaning clearly attached to a particular word or
phrase, as opposed to meaning that can be conveyed by other aspects of audio
speech than words, such as by tone of voice.

  * •

Perlocutionary refers to the effect that an utterance has on the interlocutor
or listener (as opposed to the locutionary or illocutionary aspects of the
same utterance). For example, if Interlocutor A says "it would be a shame if
something happened to your family" to Interlocutor B, the utterance has a
locutionary aspect (its literal meaning; i.e. something could happen to your
family, and that would be bad), an illocutionary aspect (the thinly veiled
threat against Interlocutor B’s family; i.e. if you don’t do as I say,
something bad will happen to your family), and a perlocutionary effect (the
fear it causes to Interlocutor B).

#### What qualifies as toxicity for this task?

The notion of toxicity can be elusive because of its inherent subjectivity;
i.e. it greatly depends on the lived experience of the person who is on the
receiving end of an utterance. For the task at hand, we arbitrarily considered
the following cases as toxic:

  * •

Profanities include slurs and language that is regarded as obscene, repulsive,
or excessively vulgar, as well as scatological. Examples of profanities in
English include words such as shit, asshole, fucking, etc.

  * •

Hate speech is language that is used to demean, disparage, belittle, or insult
groups of people. Hate speech in English includes words and expressions such
as women are sluts, men are trash, wetbacks, towelheads, etc.

  * •

Pornographic language is language that refers to sexual acts or refers in a
vulgar way to body parts typically associated with sexuality. Examples of
pornographic language include words or expressions such as suck my dick,
cumshot, etc.

  * •

Physical violence or bullying language is language that is used to bully,
threaten, silence individuals. Examples of such language include words or
expression such as son of a bitch, shut the fuck up, retard, etc.

#### What does not qualify as toxicity for this task?

Annotators were instructed to refrain from labeling as toxic the following
types of language:

  * •

Common innocuous slang is language that is particularly colloquial but isn’t
offensive; e.g., using the word cops to refer to police officers.

  * •

Appearance of hate shows up as language that conveys hate but isn’t directed
to other human beings; e.g., I hate this movie!

  * •

Appearance of obscenity shows up as language that sounds similar to
profanities or pornographic language but isn’t directed towards people; e.g.,
school sucks!

To complete the task, annotators were asked to answer the below questions:

  1. 1.

Does the utterance contain toxicity?

  2. 2.

If you answered positively to Question 1, is the toxicity related to:

    1. (a)

Specific words or phrases? If so, please specify the word or phrase.

    2. (b)

Specific perlocutionary effects? If so, please specify which effect among the
following options:

       * •

aggressive raising of the voice

       * •

aggressive tone

       * •

(veiled) threat

To help annotators reach a faster determination as to which words or phrases
could qualify as toxic in this exercise, we also pointed them to the publicly
available Toxicity-200333https://tinyurl.com/NLLB200TWL word list repository.

##  4 MuTox DataSet Description

MuTox is composed of 21 languages. Two languages, English and Spanish, contain
a larger amount of annotated data (20k utterances/sentences each language),
while the rest of languages444Bengali, Dutch, French, German, Hindi,
Indonesian, Italian, Japanese, Korean, Mandarin Chinese, Arabic, Portuguese,
Russian, Swahili, Tagalog, Thai, Turkish, Urdu, Vietnamese, contain a smaller
amount of annotated data (4k utterances/sentences each language).

###  4.1 Annotation for English and Spanish

#### Preliminary selection.

We used speech transcriptions of datasets and a text toxicity classifier. We
use mainly highly multilingual datasets: CommonVoice
555https://commonvoice.mozilla.org/ and SeamlessAlign Seamless Communication
et al. (2023b), in addition to the English/Spanish SeamlessAlignExpressive
dataset, presented in Section 2. In these datasets we use the text-based
toxicity classifier detoxify, presented in section 2, to detect toxicity in
the transcribed text.

We screen audio files by length between 2 and 8 seconds for reasons that
relate to both semantic and cognitive loads. On the one hand, annotators
reported that it was particularly difficult to assert the meaning of very
short utterances (under 2 seconds). On the other hand, longer utterances (over
8 seconds) can have too much cognitive load or too much information to
annotate. For pre-selecting toxicity samples, we perform sampling across
detoxify toxicity categories.For pre-selecting clean samples, we use cases
where all toxicity scores fall below 0.5.

#### Data statistics.

Annotation results are reported in Table 1. The percentages of confirmed toxic
samples are 16 % and 19% for English and Spanish, respectively. Figure 1
reports the types of toxicity for each language using annotated categories.The
proportion of toxicity increases with the toxicity quantile (see Figure 5 in
appendix A) . The correlation coefficient between the toxicity in the
annotation with detoxify threshold is 0.6.

| English | Spanish
---|---|---
| utterances | hours | utterances | hours
Total | 20K | 21 | 20K | 22.2
Cannot say | 547 | 0.52 | 391 | 0.41
No-Toxicity | 16210 | 17.1 | 15709 | 17.5
Toxicity | 3243 | 3.42 | 3900 | 4.24
Table 1: Results on toxicity annotated results for English and Spanish. Figure
1: Amount of toxicity per toxic category proposed in this paper.

#### Dataset splits.

The final MuTox dataset includes the 20k annotated English and Spanish corpora
from Table 1. We supplement this with the English CommonVoice annotated data
released in Detoxy Ghosh et al. (2021). We split the data in subsets of
training, dev, devtest and 3k test with stratified samples of types of
toxicity and corpora. Detailed statistics are given in Table 2.

Subset | Language | Modality | Dataset | Size | Toxicity
---|---|---|---|---|---
Train | Eng | Speech | MuTox | 13617 | 2270
Detoxy | 9818 | 2455
Text | Jigsaw | 21924 | 2928
Spa | Speech | MuTox | 13726 | 2730
| HP | 1500 | 20-360
Devtest | Eng | Speech | MuTox | 973 | 162
Spa | 981 | 195
HP | 250 | 5-60
Devtest | Eng | Speech | MuTox | 1945 | 324
Spa | 1960 | 390
HP | 750 | 10-180
Test | Eng | Speech | MuTox | 2918 | 486
Spa | 2918 | 486
HP | 1140-1480 | 15-362
Table 2: Audio speech utterances specified by dataset subset. HP languages
footnote. MuTox is our new labelled data that has been annotated in this work.
Additionally, we used data from Detoxy Ghosh et al. (2021) and JigSaw Kaggle
(2018).

###  4.2 Annotation for 19 additional languages

#### Preliminary selection.

Based on annotated data in English and Spanish (section 4.1), we devise new
criteria to select 4k sentences across 19 extra languages which we consider
high-priority (HP) for the context of this work and related previous projects
Seamless Communication et al. (2023a). The purpose of this extension is to
form a highly multilingual benchmark for audio-based toxicity classification.

Languages. Bengali, Dutch, French, German, Hindi, Indonesian, Italian,
Japanese, Korean, Mandarin Chinese, Modern Standard Arabic, Portuguese,
Russian, Swahili, Tagalog, Thai, Turkish, Urdu, Vietnamese.

Classifiers. We use a text toxicity classifier that covers all these languages
(etox) in combination with the English toxicity classifier used in section 4.1
(detoxify).

Datasets. We use SeamlessAlign which has parallel data eng–X for all languages
of interest.

Methodology. We aim at finding 2,500 sentences with the n-largest toxicity
scores, and randomly sample the dataset to complete the 4k set. Motivated by
results from section 4.1, sentences are selected as follows:

We include all the samples positive per etox and detoxify scores higher than
0.8. This results in a small number of samples, ranging from 502 (deu) to less
than 10 (hin, tgl, cmn) Then, we include etox detections (ignoring the
detoxify threshold) but allow only for a maximum of 200 sentences for each
etox token and add a maximum of 1,000 etox sentences. We set these maxima so
as to ensure toxicity diversity in the selection. Finally, we include samples
detected with detoxify: we use detoxify in Italian, Portuguese, Turkish,
Russian and French (which are supported by the tool). For the rest of non-
supported languages, we resort to English detoxify in the parallel text. [This
method is supported as follows: on the corpus from section 4.1, we have a
total toxicity of 0.25 in Spanish when using detoxify in English with a
threshold of 0.8].

Figure 2: Toxicity selection distribution per language (x-axis) and for each
method: detoxify, etox and their intersection.

#### Dataset statistics and splits.

Figure 2 shows the results of this selection, grouped by the methodology used
(1 - intersection, 2 - etox, 3 - detoxify). Table 2 shows the dataset splits.
The final dataset of 19 languages includes 120 hours.

##  5 MuTox audio-based toxicity Classifier

#### Methodology: MuTox Classifier.

We feed our toxicity classifier, MuTox, with both audio-based and text-based
toxicity-labeled data. The audio-based toxicity classifier depicted on 3
follows a simple architecture consisting of an encoder, turning input text or
audio speech into a fixed-size representation vector, and a binary classifier
composed of three simple feed-forward layers.

Figure 3: MuTox toxicity classifier diagram

#### MuTox Implementation.

We use multimodal and multilingual SONAR encoders from Duquenne et al. (2023).
This choice is motivated by the broad language coverage at the time of this
study (English, Spanish, and HP languages) and the zero-shot capabilities of
SONAR: once trained on a set of languages, the classifier head can be used on
top of any compatible SONAR encoder handling another language. For the
classifier, we use variable input sizes for the 3 feedforward layers (1024,
512, and 128). We train with a Binary Cross Entropy loss with logits and Adam
optimizer with an initial learning rate of 0.001. In order to compare zero-
shot (ZS) vs supervised performance, we train a first classifier with English
and Spanish training data only and test on HP languages. We then train a
second classifier on all training data available and report results on the
same HP languages. Results are listed in Table 2. Throughout our experiments,
we use Whisper-Large-v2 to generate transcriptions.

##  6 Experiments and Results

#### Experimental Framework.

We evaluate MuTox (text and speech) on Devtest and Test from Table 2. We
compare the performance against ASR-detoxify (hereinafter, detoxify for
simplicity) using available tool1 and ASR-etox (hereinafter, etox) using
available tool2. ASR of the speech is done with Whisper-Large-v2. We report
Area Under the Curve (AUC), Precision and Recall across toxicity detection.

#### Correlation accross classifiers

We computed Pearson correlation across classifiers see results in Table 3. To
compute the correlations we computed logarithm to MuTox outputs. We observe
that correlation is higher between MuTox and detoxify than MuTox and etox or
even detoxify and etox. In future work, we would like to do a manual analysis
of the differences across classifiers.

| ASR-MuTox | MuTox | detoxify | etox
---|---|---|---|---
ASR-MuTox | 1 | 0.75 | 0.51 | 0.13
MuTox | 0.75 | 1 | 0.49 | 0.07
detoxify | 0.51 | 0.49 | 1 | 0.26
etox | 0.13 | 0.07 | 0,26 | 1
Table 3: Pearson correlation accross toxicity classifiers.

#### Improvements of MuTox compared to one of the Strongest Quality Text-based
Toxicity Classifier

Table 4 compares MuTox and Detoxify (for available languages) in terms of AUC.
When comparing the 8 languages covered by detoxify, MuTox trained on all
languages shows a slight improvement over Detoxify of 1%. MuTox however,
scales to 10 times more languages than detoxify.

#### Improvements of MuTox compared to the Text-based Toxicity Classifier with
the Largest Coverage

Table 5 compares MuTox and etox in terms of recall at fixed precision. ASR-
MuTox with a fixed precision of $max(\textsc{ASR-ETOX},0.3)$ (meaning 0.3
average precision) improves recall by almost 30% on average.

#### MuTox Configurations

Supervised MuTox improves zero-shot MuTox by more than 7% on average. When
comparing MuTox and ASR-MuTox averaging over 21 languages, results are
comparable for supervised training. While it is unclear why MuTox vs. ASR-
MuTox show different results depending on the language, we could hypothesize
that the imbalances in the complexities of pronunciation/writing in various
languages lead to variations in transcription quality. ASR-MuTox recall (at
fixed precision) is higher than when using MuTox directly on audio speech (by
more than 8%). Note that zero-shot MuTox outperforms on average zero-shot ASR-
MuTox. On test, we get average 0.36 recall at fixed precision compared to
0.27, and 0.76 AUC compared to 0.73.

lang | Devtest (D) | Test (T) | MuToxZS | ASR-MuToxZS | MuTox | ASR-MuTox | detoxify
---|---|---|---|---|---|---|---
| Size | Tox | Size | Tox | D | T | D | T | D | T | D | T | D | T
eng | 1945 | 325 | 2918 | 486 | - | - | - | - | 0.60 | 0.64 | 0.74 | 0.76 | 0.68 | 0.71
spa | 1960 | 390 | 2941 | 585 | - | - | - | - | 0.63 | 0.67 | 0.73 | 0.73 | 0.69 | 0.71
arb | 720 | 63 | 1315 | 125 | 0.73 | 0.77 | 0.76 | 0.79 | 0.83 | 0.83 | 0.83 | 0.86 | - | -
ben | 738 | 20 | 1436 | 38 | 0.88 | 0.87 | 0.88 | 0.81 | 0.91 | 0.89 | 0.88 | 0.86 | - | -
cmn | 744 | 70 | 1347 | 140 | 0.78 | 0.79 | 0.76 | 0.77 | 0.76 | 0.81 | 0.78 | 0.77 | - | -
deu | 744 | 182 | 1124 | 362 | 0.77 | 0.81 | 0.78 | 0.79 | 0.84 | 0.87 | 0.85 | 0.86 | - | -
fra | 738 | 62 | 1353 | 124 | 0.80 | 0.80 | 0.76 | 0.78 | 0.79 | 0.83 | 0.79 | 0.80 | 0.79 | 0.83
hin | 699 | 82 | 1233 | 166 | 0.78 | 0.77 | 0.77 | 0.79 | 0.81 | 0.84 | 0.81 | 0.85 | - | -
ind | 745 | 72 | 1347 | 143 | 0.70 | 0.73 | 0.67 | 0.69 | 0.72 | 0.77 | 0.75 | 0.78 | - | -
ita | 693 | 98 | 1188 | 197 | 0.61 | 0.60 | 0.63 | 0.60 | 0.63 | 0.63 | 0.609 | 0.64 | 0.79 | 0.63
kor | 747 | 8 | 1478 | 16 | 0.71 | 0.67 | 0.80 | 0.63 | 0.80 | 0.77 | 0.79 | 0.64 | - | -
nld | 729 | 87 | 1284 | 174 | 0.83 | 0.83 | 0.70 | 0.72 | 0.77 | 0.87 | 0.86 | 0.77 | - | -
por | 724 | 109 | 1231 | 218 | 0.78 | 0.76 | 0.80 | 0.78 | 0.81 | 0.77 | 0.80 | 0.79 | 0.79 | 0.83
rus | 741 | 81 | 1320 | 161 | 0.79 | 0.76 | 0.80 | 0.82 | 0.86 | 0.79 | 0.86 | 0.85 | 0.84 | 0.81
swh | 729 | 45 | 1369 | 89 | 0.69 | 0.69 | 0.66 | 0.66 | 0.72 | 0.70 | 0.73 | 0.68 | - | -
tgl | 736 | 43 | 1385 | 88 | 0.74 | 0.73 | 0.69 | 0.70 | 0.82 | 0.82 | 0.84 | 0.78 | - | -
tha | 747 | 7 | 1480 | 15 | 0.80 | 0.76 | 0.83 | 0.66 | 0.88 | 0.85 | 0.79 | 0.78 | - | -
tur | 740 | 53 | 1373 | 107 | 0.75 | 0.74 | 0.78 | 0.73 | 0.82 | 0.81 | 0.79 | 0.80 | 0.80 | 0.82
vie | 738 | 93 | 1292 | 185 | 0.78 | 0.81 | 0.75 | 0.76 | 0.81 | 0.83 | 0.82 | 0.80 | - | -
avg8 | - | - | - | - | - | - | - | - | 0.73 | 0.74 | 0.77 | 0.77 | 0.75 | 0.76
avg | - | - | - | - | - | - | - | - | 0.79 | 0.79 | 0.80 | 0.78 | - | -
Table 4: Toxicity detection AUC results of MuTox vs detoxify. We show different MuTox configurations: ZS, trained only with English and Spanish; supervised, trained on English, Spanish, and HP languages; in audio speech (MuTox) or text (ASR-MuTox). The best results are bolded. lang | etox | MuTox-ZS | ASR-MuTox-ZS | MuTox | ASR-MuTox
---|---|---|---|---|---
Devtest | Test | Devtest | Test | Devtest | Test | Devtest | Test | Devtest | Test
| Prec | Rec | Prec | Rec | Recall (Rec)
eng | 0.39 | 0.30 | 0.40 | 0.31 | - | - | - | - | 0.61 | 0.18 | 0.75 | 0.49
spa | 0.40 | 0.33 | 0.41 | 0.33 | - | - | - | - | 0.64 | 0.19 | 0.73 | 0.44
arb | 0.09 | 0.02 | 0.18 | 0.03 | 0.16 | 0.26 | 0.18 | 0.15 | 0.57 | 0.58 | 0.40 | 0.64
ben | 0.02 | 0.02 | 0.01 | 0.02 | 0.35 | 0.11 | 0.10 | 0.03 | 0.55 | 0.40 | 0.15 | 0.37
cmn | 0.00 | 0.00 | 0.00 | 0.00 | 0.14 | 0.37 | 0.27 | 0.17 | 0.36 | 0.32 | 0.23 | 0.13
deu | 0.42 | 0.36 | 0.43 | 0.37 | 0.97 | 0.79 | 0.95 | 0.80 | 0.91 | 0.91 | 0.88 | 0.90
fra | 0.11 | 0.42 | 0.10 | 0.40 | 0.26 | 0.31 | 0.08 | 0.14 | 0.26 | 0.62 | 0.57 | 0.58
hin | 0.18 | 0.02 | 0.09 | 0.01 | 0.56 | 0.36 | 0.54 | 0.33 | 0.63 | 0.77 | 0.71 | 0.86
ind | 0.10 | 0.22 | 0.12 | 0.30 | 0.18 | 0.19 | 0.07 | 0.18 | 0.44 | 0.43 | 0.22 | 0.39
ita | 0.20 | 0.38 | 0.18 | 0.32 | 0.10 | - | 0.28 | 0.18 | 0.10 | 0.12 | 0.35 | 0.33
kor | 0.02 | 0.13 | 0.00 | 0.02 | - | - | - | - | - | - | - | -
nld | 0.08 | 0.15 | 0.06 | 0.11 | 0.77 | 0.73 | 0.21 | 0.21 | 0.79 | 0.88 | 0.62 | 0.56
por | 0.29 | 0.43 | 0.28 | 0.45 | 0.68 | 0.69 | 0.75 | 0.77 | 0.74 | 0.67 | 0.88 | 0.75
rus | 0.17 | 0.44 | 0.18 | 0.46 | 0.54 | 0.42 | 0.47 | 0.52 | 0.78 | 0.43 | 0.77 | 0.69
swh | 0.08 | 0.18 | 0.07 | 0.17 | 0.04 | 0.02 | - | 0.06 | 0.02 | - | - | -
tgl | 0.27 | 0.09 | 0.14 | 0.04 | 0.19 | 0.05 | 0.05 | 0.06 | 0.35 | 0.34 | 0.30 | -
tha | 0.00 | 0.00 | 0.00 | 0.00 | - | - | 0.14 | 0.07 | - | - | 0.14 | -
tur | 0.06 | 0.08 | 0.10 | 0.13 | 0.13 | - | 0.15 | 0.12 | 0.34 | 0.37 | 0.43 | 0.47
vie | 0.12 | 0.16 | 0.10 | 0.16 | 0.54 | 0.69 | 0.46 | 0.61 | 0.74 | 0.77 | 0.71 | 0.60
avg | 0.16 | 0.20 | 0.15 | 0.19 | - | - | - | - | 0.46 | 0.50 | 0.49 | 0.55
Table 5: Toxicity detection precision and recall results. MuTox recall at the
precision of $max(\textsc{etox}-precision,0.3)$ vs. etox. The best results are
bolded..

#### Performance across toxicity categories

Figure 4 reports the model performances by toxicity categories. We note that
detoxify and MuTox have less variance (<0.001) across categories than etox
(>0.01). etox performs well on Profanities (recall>0.8), since those can
easily be identified with explicit wordlists. It struggles however with more
complex/implicit types of toxicity such as Hate Speech or Physical Violence
(recall shows to be below 0.2 for these).

Figure 4: Recall per toxicity category at fixed precision of
$max(\textsc{etox},0.3)$.

##  7 Conclusions

In this paper, we introduced MuTox, a highly multilingual dataset and a
massive multilingual toxicity detector that lays the groundwork for the
largely unexplored task of multilingual audio-based toxicity detection. Our
MuTox dataset enables benchmarking of multilingual audio-based toxicity
detection across 21 languages. When compared to cascade tools (speech
recognition plus text-based toxicity classifiers), our MuTox classifier
demonstrates superior performance on average for all evaluated tasks.

Data666https://github.com/facebookresearch/seamless_communication/tree/main/docs/rai/mutox_README.md
and
code777https://github.com/facebookresearch/seamless_communication/tree/main/src/seamless_communication/cli/toxicity/mutox/README.md
are freely available for further research and development. In future work, we
aim to compare MuTox’s performance on the task of added toxicity and plan to
extend MuTox to more languages. We also intend to use it to analyze and
improve ETOX wordlists, as detailed in the appendix B.. This ongoing work will
continue to advance the field of multilingual audio-based toxicity detection.

## Ethical Considerations and Limitations

#### Annotators.

Annotations were provided by professional annotators, who were informed of the
nature of the content to be annotated and were given the opportunity to opt
out.

#### Bias in pre-selected sentences.

Our dataset may be biased towards text-based toxicity because we pre-selected
annotations using this criterion. The use of different text classifiers, as
well as a variety of thresholds, mitigates this potential bias.

#### Unintended bias.

Our evaluation does not cover unintended
biases888https://www.kaggle.com/competitions/jigsaw-unintended-bias-in-
toxicity-classification/data, which we intend to cover in future work.

## References

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  * Kaggle (2018) Kaggle. 2018.  Toxic comment classification challenge.  https://www.kaggle.com/c/jigsaw-toxic-comment-
classification-challenge.  Accessed: 2022-05-03.

  * NLLB Team et al. (2022) NLLB Team, Marta R. Costa-jussà, James Cross, Onur Çelebi, Maha Elbayad, Kenneth Heafield, Kevin Heffernan, Elahe Kalbassi, Janice Lam, Daniel Licht, Jean Maillard, Anna Sun, Skyler Wang, Guillaume Wenzek, Al Youngblood, Bapi Akula, Loic Barrault, Gabriel Mejia-Gonzalez, Prangthip Hansanti, John Hoffman, Semarley Jarrett, Kaushik Ram Sadagopan, Dirk Rowe, Shannon Spruit, Chau Tran, Pierre Andrews, Necip Fazil Ayan, Shruti Bhosale, Sergey Edunov, Angela Fan, Cynthia Gao, Vedanuj Goswami, Francisco Guzmán, Philipp Koehn, Alexandre Mourachko, Christophe Ropers, Safiyyah Saleem, Holger Schwenk, and Jeff Wang. 2022.  No language left behind: Scaling human-centered machine translation.
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  * Seamless Communication et al. (2023a) Seamless Communication, Loïc Barrault, Yu-An Chung, Mariano Coria Meglioli, David Dale, Ning Dong, Paul-Ambroise Duquenne, Hady Elsahar, Hongyu Gong, Kevin Heffernan, John Hoffman, Christopher Klaiber, Pengwei Li, Daniel Licht, Jean Maillard, Alice Rakotoarison, Kaushik Ram Sadagopan, Guillaume Wenzek, Ethan Ye, Bapi Akula, Peng-Jen Chen, Naji El Hachem, Brian Ellis, Gabriel Mejia Gonzalez, Justin Haaheim, Prangthip Hansanti, Russ Howes, Bernie Huang, Min-Jae Hwang, Hirofumi Inaguma, Somya Jain, Elahe Kalbassi, Amanda Kallet, Ilia Kulikov, Janice Lam, Daniel Li, Xutai Ma, Ruslan Mavlyutov, Benjamin Peloquin, Mohamed Ramadan, Abinesh Ramakrishnan, Anna Sun, Kevin Tran, Tuan Tran, Igor Tufanov, Vish Vogeti, Carleigh Wood, Yilin Yang, Bokai Yu, Pierre Andrews, Can Balioglu, Marta R. Costa-jussà, Onur Celebi, Maha Elbayad, Cynthia Gao, Francisco Guzmán, Justine Kao, Ann Lee, Alexandre Mourachko, Juan Pino, Sravya Popuri, Christophe Ropers, Safiyyah Saleem, Holger Schwenk, Paden Tomasello, Changhan Wang, Jeff Wang, and Skyler Wang. 2023a.  Seamlessm4t-massively multilingual & multimodal machine translation.
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  * Thewlis et al. (2021) James Thewlis, Sasha Haco, and Laura Hanu. 2021.  How ai is learning to identify toxic online content.
  * Yousefi and Emmanouilidou (2021) Midia Yousefi and Dimitra Emmanouilidou. 2021.  Audio-based toxic language classification using self-attentive convolutional neural network.  In _29th European Signal Processing Conference, EUSIPCO 2021, Dublin, Ireland, August 23-27, 2021_ , pages 11–15. IEEE.

##  Appendix A Annotations Details

Figure 5 shows the percentage of toxicity obtained with the annotation
(y-axis) by the quantile of toxicity (x-axis) in the text data.

Figure 5: Percentage (y-axis) of toxicity in the audio speech dataset in
English (top) and in Spanish (bottom) per toxicity quantile (x-axis) in the
text toxicity classification.

##  Appendix B Wordlists analysis.

We report an analysis on the toxic words detected with etox in order to
understand the limitations of the Toxicity-200 word-lists.

For English we have a total of 110 different toxic tokens detected and 59
tokens show reasonable precision (>0.4). On the one hand, the worst performing
tokens are insults such as stupid* and fool*, these insults are not so harsh
and probably some times are not considered non-toxic by native speakers. These
show a high output, but very low precision and hence are negatively affecting
etox overall’s performance. On the other hand, some of the best performing
tokens are variations of fuck* and shit*. These tokens show a high output and
precision (>0.8), being responsable for a considerable share of recall
(>0.15). Slurs tend to have a high precision but low recall.

For Spanish we have a total of 187 different toxic tokens detected and 110
tokens show reasonable precision (>0.4). The worst performing tokens are
insults such as tonto* and maldito* and some of the best performing tokens are
are m*erda and variations of j*der. The recall is much more distributed
compared to English, with several terms having a precision above 0.4.

Figure 6: ETOX’s tokens performance for our English Dataset (top) and Spanish
Dataset (bottom). Vertical axis representing the precision and Horizontal axis
the total output.

This analysis could be further extended to HP languages.

##  Appendix C Data Card - MuTox

Dataset Summary999We use a template for this data card
https://huggingface.co/docs/datasets/v1.12.0/dataset$\\_$card.html • MuTox is
a speech and text dataset on 21 languages with toxicity labels. How to use the
data • You can access links to the data in the README at
https://github.com/facebookresearch/seamless_communication/tree/main/docs/rai/mutox_README.md
Data Structure • Languages: This version of MuTox cover a total of 21
languages: English, Spanish, Arabic, Bengali, Mandarin Chinese, Dutch, French,
German, Hindi, Indonesian, Italian, Japanese, Korean, Portuguese, Russian,
Swahili, Tagalog, Thai, Turkish, Urdu, Vietnamese • Fields available: Speech
source, Text source, Toxicity labels Dataset Splits • We provide training,
dev, devtest and test splits as defined in Table 2. Dataset Creation •
Preprocessing steps: On speech, we pre-filter audios by length between 2 and 8
seconds for convenience. On text, we pre-selected data based on all criteria
described in section 3. • Annotations Toxicity labels where produced based on
annotation guidelines from section 3.  • Personal and Sensitive Information:
Not applicable Considerations for Using the Data • Social Impact of Dataset:
The dataset is built to increase toxicity detection quality in speech and text
natural language applications. Additional Information • Licensing Information
We are releasing training, dev and devtest splits under the terms of MIT
License • Citation Information
Marta R. Costa-jussà, Mariano Coria Meglioli, Pierre Andrews, David Dale,
Prangthip Hansanti, Elahe Kalbassi, Alex Mourachko, Christophe Ropers,
Carleigh Wood, MuTox: Universal MUltilingual Audio-based TOXicity Dataset and
Zero-shot Detector, arxiv, 2024

##  Appendix D Model Card - MuTox

Model Details • Person or organization developing model: Developed by FAIR at
Meta • Model date: November 30, 2023 • License: CC-BY-NC 4.0 Intended Use •
Primary intended uses: Automatic detection of toxicity for speech and text. •
Information on how to use the model can be found in the
Seamless$\\_$communication repository:
https://github.com/facebookresearch/seamless_communication/tree/main/src/seamless_communication/toxicity
• Primary intended users: Primary users are researchers and speech translation
research community.  Metrics • Model performance measures: AUC, Precision and
Recall Data • Training and Evaluation data: Detoxy Ghosh et al. (2021), JigSaw
Kaggle (2018) and MuTox Corpus, see Table 2 and MuTox data card in appendix B.
Ethical Considerations • The training data used for model development were
mined from various publicly available sources on the web. Although we invested
heavily in data cleaning, personally identifiable information may not be
entirely eliminated. However, as the model is not generative, it is unlikely
that such information could be recovered from the model.  • We did not
specifically study possible biases of the model. Caveats and Recommendations •
Limitations: Researchers should consider extending data to other languages and
more types of toxicity when using the model in a research application.
Citation Information Marta R. Costa-jussà, Mariano Coria Meglioli, Pierre
Andrews, David Dale, Prangthip Hansanti, Elahe Kalbassi, Alex Mourachko,
Christophe Ropers, Carleigh Wood, MuTox: Universal MUltilingual Audio-based
TOXicity Dataset and Zero-shot Detector, arxiv, 2024


# Constraining Scale Dependent Growth with Redshift Surveys

Mikhail Denissenya1, Eric V. Linder1,2 1Energetic Cosmos Laboratory,
Nazarbayev University, Nur-Sultan, Qazaqstan 010000
2Berkeley Center for Cosmological Physics & Berkeley Lab, University of
California, Berkeley, CA 94720, USA

###### Abstract

Ongoing and future redshift surveys have the capability to measure the growth
rate of large scale structure at the percent level over a broad range of
redshifts, tightly constraining cosmological parameters. Beyond general
relativity, however, the growth rate in the linear density perturbation regime
can be not only redshift dependent but scale dependent, revealing important
clues to modified gravity. We demonstrate that a fully model independent
approach of binning the gravitational strength $G_{\rm eff}(k,z)$ matches
scalar-tensor results for the growth rate $f\sigma_{8}(k,z)$ to
$0.02\%$–$0.27\%$ rms accuracy. For data of the quality of the Dark Energy
Spectroscopic Instrument (DESI) we find the bin values can be constrained to
1.4%–28%. We also explore the general scalar-tensor form, constraining the
amplitude and past and future scalaron mass/shape parameters. Perhaps most
interesting is the strong complementarity of low redshift peculiar velocity
data with DESI-like redshift space distortion measurements, enabling
improvements up to a factor 6–7 on 2D joint confidence contour areas. Finally,
we quantify some issues with gravity parametrizations that do not include all
the key physics.

##  I Introduction

Cosmic growth of large scale structure is a competition between increased
clustering amplitude driven by gravity and damped evolution due to expansion,
exacerbated by cosmic acceleration. Thus the growth, and more incisively the
growth rate, can reveal important characteristics of gravitation and dark
energy. The new generation of spectroscopic galaxy surveys measure redshift
space distortions across wide areas of sky and broad redshift ranges, enabling
precision estimates of the growth rate $f\sigma_{8}$.

Within general relativity these measurements translate directly to cosmic
expansion history constraints, but the extra freedom within modified gravity
means that redshift space distortions can serve as a critical probe of gravity
(see, e.g., [1, 2, 3, 4] for early work). Moreover, with modified gravity
there generally enters scale dependence in the growth rate, even in the linear
density perturbation regime, a further signal of deviation from Einstein
gravity. We focus here on constraining scale dependent growth, in as model
independent a manner as practical, with the quality of data likely to be
delivered by the current spectroscopic surveys such as from the Dark Energy
Spectroscopic Instrument (DESI, [5, 6]).

In Section II we investigate the evolution of the scale- and redshift-
dependent cosmic growth rate $f\sigma_{8}(k,a)$ and explore the impact of the
physics motivated scalar-tensor gravity form for the modified gravity
strength, keeping it as model independent as reasonable. Section III then
demonstrates that a wholly model independent form, using $2\times 3$ bins in
scale and redshift, can provide an excellent approximation to the full form.
We project constraints in Section IV for what DESI-quality data could enable
to test gravity through scale dependent growth in these two approaches.
Leverage from peculiar velocity low redshift observations is examined as well.
Section V studies a coarser, but easier to be alerted to distinctions from
general relativity, $2\times 2$ bin approach. In Section VI we summarize and
conclude.

##  II Scale Dependent Growth

Galaxy redshift surveys use redshift space distortions to determine the growth
rate factor

$f\sigma_{8}(a)=fD\,\frac{\sigma_{8,0}}{D_{0}}=\frac{dD}{d\ln
a}\,\frac{\sigma_{8,0}}{D_{0}}\ ,$ (1)

where $D$ is the linear perturbation growth factor, $a$ is the expansion
factor, and $\sigma_{8}$ the mass fluctuation amplitude; subscript 0 denotes
the value today. Unlike for the logarithmic growth rate $f=d\ln D/d\ln a$, the
evolution equation for $D$ is linear:

$D^{\prime\prime}+D^{\prime}\left[2+\frac{1}{2}\left(\ln
H^{2}\right)^{\prime}\right]-\frac{3}{2}DG_{m}\Omega_{m}(a)=0\ ,$ (2)

where prime denotes a derivative with respect to $\ln a$.

It is convenient to break this second order equation into two coupled first
order equations,

$\displaystyle D^{\prime}$ $\displaystyle=$ $\displaystyle(fD)$ (3)
$\displaystyle(fD)^{\prime}$ $\displaystyle=$
$\displaystyle-(fD)\left[2+\frac{1}{2}\left(\ln
H^{2}\right)^{\prime}\right]+\frac{3}{2}DG_{m}\Omega_{m}(a)\ .$ (4)

Here $H(a)$ is the Hubble parameter $\dot{a}/a$, and $\Omega_{m}(a)$ is the
matter density in units of the critical density. The factor $G_{m}$ is the
gravitational coupling strength of matter, in units of Newton’s constant. In
general relativity it is simply one.

However, in modified gravity $G_{m}$ can differ from one, and be both
expansion factor (redshift) dependent and scale (wavemode) dependent. Since we
work in the linear regime, linear perturbations at one wavenumber $k$ are
independent of those at another. In general then we have $G_{m}(k,a)$,
determining $f\sigma_{8}(k,a)$.

###  II.1 Modified Gravity

One could adopt a particular theory of modified gravity, with a certain
functional form, and certain parameters within that form – e.g. $f(R)$ gravity
of the Hu-Sawicki form with parameter $n=4$ – but the constraints derived from
the data will then apply only to that specific case. We take a more model
independent approach, using that in scalar-tensor theories the gravitational
coupling to matter can be fairly generically written as [7, 8, 9, 10, 11, 12,
13, 14]

$G_{m}=\frac{1+c\,\left(k/[aM(a)]\right)^{2}}{1+\left(k/[aM(a)]\right)^{2}}\
,$ (5)

(in the quasistatic regime, where precision observations can be made). We take
the scalaron mass to be given by

$M(a)=M_{a}a^{-3}+M_{\rm dS}\ ,$ (6)

which fits many models reasonably [15] (also see [16]), and guarantees a high
redshift universe looking like that in general relativity. Since in general
relativity $G_{m}=1$, we define $\delta G=G_{m}-1$. We refer to $c$ as the
amplitude parameter and $M_{a}$ and $M_{\rm dS}$ as the shape parameters,
since they govern the evolution of scale dependence. Note that at high
redshift the scalaron mass is large, $M\approx M_{a}a^{-3}$, today
$M(a=1)=M_{a}+M_{\rm dS}$, and in the future the scalaron mass freezes as the
universe approaches a de Sitter state, $M(a)\to M_{\rm dS}$. Since today the
expansion is not too far from the de Sitter state, i.e. $\Omega_{\rm
de}\approx 0.7$, then we expect $M(a=1)\lesssim 2M_{\rm dS}$, i.e.
$M_{a}\lesssim M_{\rm dS}$.

Consider a fiducial model of $c=4/3$, $M_{a}=0.05$, $M_{\rm dS}=0.05$. (All
wavenumbers and masses are in $h$/Mpc units.) The value $c=4/3$ matches $f(R)$
gravity [11], and the value $M(a=1)=M_{a}+M_{\rm dS}=0.1$ gives a value for
the Compton wavelength parameter [17] $B_{0}=2[H_{0}/M(a=1)]^{2}\approx
2.2\times 10^{-5}$. The exact translation of $B_{0}$ to the $f(R)$ parameter
$f_{R0}$ is model dependent, but generally $f_{R0}\approx-B_{0}/(1-4)$. Thus
the fiducial model has $f_{R0}\approx-10^{-5}$, toward the high end of what is
allowed by current data, designed to stress test the ability of our binned
approach to accurately fit the growth data. Finally note that the power
$a^{-3}$ in Eq. (6) also stress tests our approach in that this is the
shallowest power allowed: one requires $M(a\ll 1)\sim a^{\leq-3}$ [11, 18] to
obtain the standard matter dominated Friedmann expansion equation behavior.
Using the limit $M(a\ll 1)\sim a^{-3}$ keeps modifications stronger at earlier
times and so makes fitting growth accurately with a small number of $\delta G$
bins more difficult.

Figure 1:  Modified gravitational strength deviations from GR are plotted as
$\delta G$ vs $k$, at $a=0.8$ (left panel), and $\delta G$ vs $a$ at $k=0.125$
(right panel). The fiducial case of $c=4/3$, $M_{a}=M_{\rm dS}=0.05$ (bold
black curve) and several variations as labeled by $(c,M_{a},M_{\rm dS})$ are
shown, as well as an unmotivated power law rather than Padé polynomial case
(dashed red).

Figure 1 shows $\delta G(k)$ for $a=0.8$, i.e. in the recent universe where
the deviation from general relativity (GR) is larger, and $\delta G(a)$ for
$k=0.125$, i.e. at a higher wavenumber where the deviation is larger. We
illustrate the effects of changing the parameters one by one. Increasing the
amplitude parameter $c$ of course scales $\delta G$ at all $k$ and $a$
(compare thin blue vs bold black curves). For the shape parameters entering
the scalaron mass, recall that large mass increases the similarity to GR.
Increasing the shape parameter $M_{a}$ greatly suppresses modifications until
recently, for example we see that $\delta G$ is three times smaller at $a=0.5$
(for $k=0.125$) for $M_{a}=0.1$ rather than $M_{a}=0.05$ (compare dot dash
purple vs bold black curves). The shape parameter $M_{\rm dS}$ changes the
late time behavior; it is not until $a\gtrsim 0.5$ that increasing $M_{\rm
dS}$ from 0.05 to 0.1 has an appreciable effect (dot dot dash green vs bold
black).

There are two other points of interest to note. First, using a form like
$G_{m}=1+ba^{s}$ is quite problematic (see also, e.g., [19]); it gives very
different behavior than the full Padé ratio of Eq. (5) – see the dashed red
curve in Fig. 1. It drastically overweights late times (and high $k$) so that
a constraint from late time data could be interpreted as an unrealistically
tight limit on modified gravity. Recall that the Padé form originates from the
Einstein field equations [7, 8, 9, 10, 12, 13, 14]; basically $G_{m}$ comes
from $F_{1}$(metric potential) = $F_{2}$(matter fields), where $F_{i}$ are
factors including space and time derivative terms, and matter fields include
the scalar field perturbations; both sides involve $k^{2}$ corrections beyond
leading order. Since $G_{m}$ arises from a ratio of the factors, using only a
single polynomial rather than a Padé form is throwing away half the physics.
This holds for any scalar-tensor theory. We can see from Fig. 1 that the
discrepancy is already significant for $a\gtrsim 0.5$ or $k\gtrsim 0.06$.

Secondly, as mentioned the scalaron mass freezes approaching a dark energy
dominated de Sitter state, $G_{m}\not\to c$ but rather gravity is eventually
restored to GR, $G_{m}\to 1$ since $aM(a)\to aM_{\rm dS}$ and the factors
multiplying the $k^{2}$ terms go to zero. The “phase space” evolution of
$G_{m}$ for various modified gravity theories is discussed and illustrated in
[15]. Whether $G_{m}$ reaches its fully modified value of $c$ before turning
around and approaching back to 1 depends on the values of $M_{a}$ and $M_{\rm
dS}$. We illustrate this in Fig. 1 with the dotted orange curve, where $M_{a}$
and $M_{\rm dS}$ are chosen to match $M(a=0.8)$ from the fiducial model
(specifically, we use $M_{a}$=0.02, $M_{\rm dS}=0.1086$). Indeed, the left
panel shows perfect overlap between those two cases, while the right panel
shows that they coincide only at (the chosen) $a=0.8$. The turnaround is
clearly seen: the dotted orange curve never exceeds $\delta G\approx 0.18$,
well short of $c-1=0.33$. All of the curves (except the non-Padé one) will
eventually turn around; it simply occurs later for smaller $M_{\rm dS}$.

###  II.2 Growth Rate

Given $G_{m}$, Eq. (5), one can then solve Eq. (4) and obtain the scale
dependent growth rate $f\sigma_{8}(k,a)$ in the linear regime. Figure 2 shows
$(f\sigma_{8}-f\sigma_{8}^{GR})/f\sigma_{8}^{GR}$ vs $a$ for various $k$, and
vs $k$ for various $a$, for the fiducial modified gravity model, in a
$\Lambda$CDM background with fractional matter density today $\Omega_{m}=0.3$.
(Recall that the maximum equation of state deviation from $-1$ for many $f(R)$
theories is $|1+w|_{\rm max}\approx B_{0}/2\ll 1$ [20], so $\Lambda$CDM is an
excellent approximation.) We normalize the amplitude to the same primordial
density amplitude as GR (as measured by the cosmic microwave background),
rather than to the present mass fluctuation amplitude $\sigma_{8,0}$ (which
would cause deviations at high redshift).

Figure 2: The fractional difference of $f\sigma_{8}$ relative to general
relativity is plotted vs $k$, for various scale factors $a$ [left panel], and
vs $a$ for various wavemodes $k$ [right panel].

The growth rate $f\sigma_{8}$ has less than 1% deviation from GR for the
fiducial model earlier than about one efold before the present, $a\lesssim
0.37$, for $k<0.15$. Lower wavenumbers deviate later and with a smaller
amplitude. Between $a=0.4$ ($z=1.5$) and $a=0.8$ ($z=0.25$), the deviation
grows by a factor of $\sim 6$. Thus redshift space distortion data over a
broad range of redshifts is highly useful. The deviation from GR in the growth
rate increases with $k$, reaching 2–10% at $k=0.15$ for $a=0.4$–0.8, the range
of near term precision redshift space distortion data. Thus we see that
percent level measurements of $f\sigma_{8}$ will be required, somewhat eased
by adding up measurements at many redshifts. It is important to remember that
$f\sigma_{8}(k,a)$ is affected by all values $G_{m}(k,a^{\prime}\leq a)$; i.e.
while modes evolve independently within the linear regime, modifications in
the gravitational strength at earlier times affect the growth rate at later
times (effectively changing the “momentum” of the growth, see [21]).

##  III Model Independent Fitting

Within scalar-tensor theories, we then have a three parameter fit
$(c,M_{a},M_{\rm dS})$ for gravity that can be used for testing GR with growth
rate data. This is a valid and useful approach, but while it is the most
physical approach without adopting a specific gravity model it does have some
drawbacks. One must solve the growth equation for each point in the parameter
space, and there is some degeneracy. High values of $M(a)$, coming from either
large $M_{a}$ or large $M_{\rm dS}$, are indistinguishable from GR (for a
Monte Carlo fitting approach one might want to adopt priors on $1/M_{a}$,
$1/M_{\rm dS}$ instead). Covariance between the parameters can only partially
be ameliorated with data over a range of $k$ and $a$. Finally, while the Padé
form is model independent within canonical scalar-tensor theories, we might
want to test the framework itself.

Therefore we will also investigate an even more model independent approach,
that of independent bin values of $G_{m}$ in $k$ and $a$. We want to keep the
number of bins small, but sufficient to both see the physics qualitatively and
reproduce the growth rate with quantitative accuracy. Note that the bins are
in $G_{m}$, giving the full, continuous function $f\sigma_{8}(k,a)$. (For some
studies using gravity binned in scale and redshift, see [22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33].)

Since we restrict ourselves to the quasilinear regime, we choose two bins in
$k$: low $k=[0.01,0.1]$ and high $k=[0.1,0.15]$. Subdividing further would
lead to increased covariance between parameters and weaker constraints. For
expansion factor $a$, we recognize that models such as $f(R)$ have very rapid
variation in their time dependence, e.g. $\delta G\sim a^{4}$ at low $k$, low
$a$. This led [34] to adopt three bins in $a$ for accuracy, and we follow this
procedure. We take bins, in redshift, of $z=[0,0.5]$, [0.5,1], and [1,3].
Again we emphasize that these are bins for $G_{m}$ and not in $f\sigma_{8}$.
Although we saw that $f\sigma_{8}$ deviates from GR by less than 1% for
$a<0.37$, we must allow $G_{m}$ to deviate earlier since those deviations
affect all later $f\sigma_{8}$, even up to $z=0$. From Fig. 1 we see that
$\delta G$ deviates by less than 1% at $a<0.25$, and set $G_{m}=1$ at
$a<0.25$. Note that while $G_{m}$ is piecewise constant in bins, as we never
take derivatives of $G_{m}$ then jumps at the bin boundaries do not cause
difficulties for the calculation; smoothing the transitions was studied in
[34] and found not to affect substantially the results.

From the $2\times 3$ binned values of $G_{m}$ we compute
$\widehat{f\sigma_{8}}(k,a)$ and compare this to $f\sigma_{8}$ from the full,
Padé functional form of $G_{m}$. Minimizing the variance of the fractional
deviation $(\widehat{f\sigma_{8}}-f\sigma_{8})/f\sigma_{8}$ determines the
optimal values for the binned $G_{m}$. Note that while each $k$ bin is
independent from the others, since the growth equation does not couple modes
in the linear regime, the evolution of $\widehat{f\sigma_{8}}$ at some $a$
does depend on $G_{m}$ at earlier $a$ as this affects $D(a)$, which has been
integrated from the high redshift, GR initial conditions to $a$.

In more detail, we recognize that observational data will not have
$f\sigma_{8}(k,a)$ at every $k$ with sufficient signal to noise (S/N) to test
gravity, but rather one must assess $f\sigma_{8}$ for modes over some range.
So as not to decrease the S/N of the data too much we choose two bins of $k$
in $f\sigma_{8}$, using $k=[0.01,0.1]$ and $k=[0.1,0.15]$. Since a deviation
$\delta G$ at some $k$ leads to a deviation in $f\sigma_{8}$ at exactly that
$k$ and no other, in the linear regime, it is convenient to take the
$f\sigma_{8}$ bins in $k$ to be the same as the $G_{m}$ bins in $k$.
Measurements would be quoted at some $k_{\rm eff}$ in each bin, which here we
will simply take as the bin center, i.e. $k_{\rm eff}=0.055$, 0.125. (One
could take into account the mode density, dependence of the power spectrum on
$k$, spectrograph fiber collision weights [35], etc. to come up with a more
sophisticated $k_{\rm eff}$ but we simply take the bin center.) In redshift,
data is usually reported in fairly narrow redshift slices, approximating a
continuous function for $f\sigma_{8}$ in terms of $a$, so we do not bin
$f\sigma_{8}$ in $a$ (while $G_{m}$ values are defined in bins in $a$).
However, the precision of redshift space distortion measurements degrades at
$z<0.2$ where there is relatively little volume (but see Sec. IV.3 where we
revisit this using peculiar velocity surveys). At high redshift, again the
precision worsens due to observational difficulties. Therefore we carry out
the optimization of $G_{m}(k,a)$ bins by minimizing the variance of deviations
in $f\sigma_{8}$ over the range $z=[0.2,1.7]$. (Recall that $G_{m}$ at $z>1.7$
still affects $f\sigma_{8}$ over this whole range.)

Figure 3 shows the fractional residuals of the fit
$(\widehat{f\sigma_{8}}-f\sigma_{8})/f\sigma_{8})$ vs $k$ for various values
of $a$, and vs $a$ for various $k$. However, what is important are the results
at the two values $k_{\rm eff}=0.055$ and 0.125 since $f\sigma_{8}(k_{\rm
eff},a)$ is what the data will actually provide. These values are marked by
the vertical lines in the left panel, and the solid curves in the right panel.
We see that over the redshift range of the data, the binned $G_{m}$ approach
succeeds in delivering accurate results, with the maximum deviation $\lesssim
0.1\%$ in the low $k$ bin, and $\lesssim 0.1\%$ in the high $k$ bin except
right around $a=0.5$ (a $G_{m}$ bin boundary) where the maximum deviation is
$\approx 0.6\%$. Changing from piecewise constant $G_{m}$ to a smoother form
would reduce this deviation, but 0.6% is still sufficiently good as we will
see the observational precision at this redshift is expected to be $\approx
2\%$ when the redshift space distortions are analyzed in two $k$ bins. The rms
deviations are even smaller, as shown in Table 1.

Range | rms | max
---|---|---
low $k$, low $a$ |  0.07% |  0.13%
low $k$, mid $a$ | 0.04% | 0.13%
low $k$, high $a$ | 0.02% | 0.04%
high $k$, low $a$ | 0.27% | 0.64%
high $k$, mid $a$ | 0.22% | 0.64%
high $k$, high $a$ | 0.06% | 0.12%

Table 1: Fractional deviations on the growth rate $f\sigma_{8}$ computed using
binned $G_{m}$ vs the exact form. Low/high $k$ refers to $k_{\rm eff}=0.055$,
0.125 respectively; low/mid/high $a$ refers to the ranges [0.37,0.5],
[0.5,0.667], [0.667,0.833]. The rms and max columns give the rms and maximum
deviations across those ranges of $a$.

Figure 3: The fractional difference of $f\sigma_{8}$ using the optimized
binned $G_{m}^{\rm bin}$ (giving $\widehat{f\sigma_{8}}$) vs using the exact
modified gravity $G_{m}$ (giving $f\sigma_{8}^{\rm Exact}$) is plotted vs $k$
for various $a$ [left panel] and vs $a$ for various $k$ [right panel]. Only
the values at $k_{\rm eff}=0.055$, 0.125 are observationally relevant (since
$f\sigma_{8}$ data cannot be divided into more than two bins without losing
too much precision). These are indicated with vertical lines in the left
panel, and solid curves in the right panel. The observationally relevant (i.e.
good data precision) range in $a$ is indicated by the unshaded region in the
right panel.

While in Fig. 3 we show $(\widehat{f\sigma_{8}}-f\sigma_{8})/f\sigma_{8}$ at
other values of $k$ than $k_{\rm eff}$, we again emphasize that the values at
$k_{\rm eff}$ are what is important when comparing with data. The $G_{m}$
values obtained from our variance minimization are: $\delta G^{\rm
bin}_{11}=0.0065$, $\delta G^{\rm bin}_{12}=0.031$, $\delta G^{\rm
bin}_{13}=0.052$, $\delta G^{\rm bin}_{21}=0.029$, $\delta G^{\rm
bin}_{22}=0.127$, $\delta G^{\rm bin}_{23}=0.158$, where $\delta G^{\rm bin}$
denotes $G_{m}^{\rm bin}-G^{GR}$ and the numerical indices give the bins of
$G_{m}$ in $k$ and $a$ respectively, from low values of $k$ or $a$ to high.
For example, $\delta G^{\rm bin}_{23}$ is the optimized binned deviation in
gravity in the $G_{m}$ bin $k=[0.1,0.15]$, $a=[0.667,1]$. As expected there is
a clear trend of strengthening gravity for increasing $k$ or $a$.

##  IV Projected Constraints

To estimate parameter constraints on the gravitational model we can employ the
information matrix formalism, where the information matrix is

$\displaystyle F_{ij}$ $\displaystyle=$ $\displaystyle\sum_{k_{\rm
eff,n}}\sum_{z_{m}}\frac{\partial f\sigma_{8}(k_{\rm eff,n},z_{m})}{\partial
p_{i}}\frac{\partial f\sigma_{8}(k_{\rm eff,n},z_{m})}{\partial p_{j}}$ (7)
$\displaystyle\qquad\times\frac{1}{[\sigma(k_{\rm
eff,n},z_{m})f\sigma_{8}(k_{\rm eff,n},z_{m})]^{2}}\ ,$ (8)

and the marginalized parameter constraints are given by the elements of
$F^{-1}$, with the uncertainty on a parameter equal to the square root of its
diagonal element, i.e. $\sigma(p_{i})=\sqrt{(F^{-1})_{ii}}$.

The set of data $f\sigma_{8}(k_{\rm eff,n},z_{m})$ are taken to follow those
projected for the Dark Energy Spectroscopic Instrument (DESI), using only the
redshift space distortion galaxy data. We include the redshift range
$z_{m}=0.25,0.35,\dots 1.65$ where precision is strong, with the uncorrelated
fractional precisions $\sigma(k_{\rm eff,n},z_{m})$ given by Tables 2.3 and
2.5 of [5]. We evaluate $f\sigma_{8}$ at $k_{\rm eff}=0.055$ and 0.125,
reflecting that the observed data must extend over a range, here respectively
$k=[0.01,0.1]$ and $k=[0.1,0.15]$, to obtain good precision. For the lower $k$
bin, we use the column in the $k_{\rm max}=0.1$ column of those tables; for
the higher $k$ bin we use $\sigma_{[0.1,0.15]}=1.5(\sigma_{k_{\rm
max}=0.2}^{-2}-\sigma_{k_{\rm max}=0.1}^{-2})^{-1/2}$, where the 1.5 factor is
a reasonable approximation for the uncertainty for the range $k=[0.1,0.15]$ vs
[0.1,0.2]. The background cosmology is taken as $\Lambda$CDM with $\Omega_{m}$
a parameter to be marginalized over.

###  IV.1 Scalar-Tensor Parameter Constraints

We first consider the general scalar-tensor form Eq. (5), so our fit
parameters are $(c,M_{a},M_{\rm dS},\Omega_{m})$, with fiducial values
$(4/3,0.05,0.05,0.3)$. Our standard analysis includes a prior
$\sigma(\Omega_{m})=0.01$ but we also examine variations of this.

Results show that the amplitude parameter $c$ cannot be constrained, with
$\sigma(c)\approx c$ even when fixing $\Omega_{m}$. This is due to strong
covariance with the shape parameters $M_{a}$ and $M_{\rm dS}$ (correlation
coefficients $0.994$ and $0.974$ respectively) – since $c$ is both redshift
and scale independent, it cannot be distinctly separated from the parameters
that do affect the redshift and scale dependence of $f\sigma_{8}$. Only when
those parameters are fixed does the uncertainty on the amplitude become
constrained, $\sigma(c)\approx 0.11$ with a 0.01 prior on $\Omega_{m}$ or
$\sigma(c)\approx 0.07$ with $\Omega_{m}$ fixed.

This is an important point: analyses that do not take into account the physics
of the form Eq. (5) for scalar-tensor theories, but rather constrain an offset
in the low $k$ (and often scale independent) limit, such as $G_{m}-1=\delta
G(z=0)(1+a^{3})$, are addressing a very different physical model for gravity.
Such constraints benefit greatly from the persistence effect: that any offset
in $G_{m}$ at any redshift influences $f\sigma_{8}$ at all later redshifts. In
contrast, the scalar-tensor form has little offset in $G_{m}$ at low $k$ at
any redshift and so it is more difficult to constrain these physical models.

To save the scalar-tensor form to some extent, we might fix $c=4/3$, the value
for $f(R)$ theories. Here we give up some model independence and are now
basically constraining $f(R)$ theories – recall that the shape parameters
$M_{a}$, $M_{\rm dS}$ are related to $B_{0}$ and $f_{R0}$ by roughly

$-f_{R0}\approx B_{0}/2\approx\left[\frac{H_{0}}{M(a=1)}\right]^{2}\approx
10^{-5}\left(\frac{0.1}{M_{a}+M_{\rm dS}}\right)^{2}\ .$ (9)

We find constraints of $\sigma(M_{a})=0.02$, $\sigma(M_{\rm dS})=0.18$,
insensitive to the $\Omega_{m}$ prior. Recall that at high redshift, the
$M_{a}$ term in Eq. (6) dominates over the $M_{\rm dS}$ term, and so is much
better constrained. The joint 68% confidence contour in the $M_{a}$–$M_{\rm
dS}$ plane is shown in Fig. 4. Note that the larger $M_{a}$ and $M_{\rm dS}$
are, the closer the observables are to the GR case.

One could propagate these uncertainties to the quantity $M_{a}+M_{\rm dS}$ to
obtain an estimation uncertainty on $f_{R0}$. However, the Gaussian
approximation of the information matrix approach is not robust here, giving
$\sigma(M_{a}+M_{\rm dS})/(M_{a}+M_{\rm dS})\gtrsim 1$. If one naively took
that the $1\sigma$ upper limit on their sum is 0.27, this would imply a lower
limit $|f_{R0}|\gtrsim 1.4\times 10^{-6}$. That is, for the given fiducial the
data could distinguish the model from GR at 68% CL. Note that the picture
would be very different if one neglected the physics of the approach to the de
Sitter state and ignored $M_{\rm dS}$. If the mass parameter were taken to be
$M(a)=M_{a}a^{-3}$ only, then $\sigma(M_{a})=\sigma(M(a=1))=0.01$ and one
would conclude that $M_{a}=0.1\pm 0.01$ (keeping the fiducial as $M(a=1)=0.1$)
and therefore $|f_{R0}|=[0.8,1.2]\times 10^{-5}$ – a very different physics
conclusion! This highlights the importance of including all the key physics in
interpreting the constraints from data.

Figure 4:  Joint 68% confidence contours for $M_{a}$ and $M_{\rm dS}$ with
fixed $c=4/3$, with a prior of 0.01 on $\Omega_{m}$ (solid curve) or fixed
$\Omega_{m}$ (dashed curve).

###  IV.2 Binned Gravity Constraints

To return to a more model independent approach to constraining gravity, we now
estimate uncertainties of the $2\times 3$ bin values of $G_{m}$, i.e. looking
for trends between low and high $k$ (larger and smaller scales) and from high
to mid to low redshift. The parameter set is now
$(G_{11},G_{12},G_{13},G_{21},G_{22},G_{23},\Omega_{m})$, and their fiducial
values are the optimized fit values as described in Sec. III.

Table 2 compiles the results. At low $k$, the effect of the modified gravity
on the observable $f\sigma_{8}$ is smaller than the precision of the data, and
it is difficult to see any distinction from GR (i.e. $G_{m}=1$). For the
$k_{\rm eff}=0.125$ bin, however, deviations from GR do start to have a
noticeable effect. Note that since deviations in the low $a$ (high redshift)
bin impact $f\sigma_{8}$ at all later redshifts, the precision on $G_{m}$ at
low $a$ is tightest. At high $a$ (low redshift), the $G_{m}$ bin extending
from $z=[0,0.5]$ only affects three data slices having reasonable precision:
those at $z=0.25$, 0.35, 0.45, and so the uncertainties on $G_{m}$ at high $a$
become larger.

Bin of $G_{m}$ | $G_{\rm bin}-1$ | $\sigma(G_{\rm bin})$ | $\sigma_{+}(G_{\rm bin})$
---|---|---|---
low $k$, low $a$ [11] |  0.006 |  0.019 |  0.018
low $k$, mid $a$ [12] | 0.031 | 0.104 | 0.103
low $k$, high $a$ [13] | 0.052 | 0.377 | 0.137
high $k$, low $a$ [21] | 0.029 | 0.018 | 0.018
high $k$, mid $a$ [22] | 0.127 | 0.094 | 0.093
high $k$, high $a$ [23] | 0.158 | 0.290 | 0.135

Table 2: Information matrix constraints on $G_{m}$ in each of the $2\times 3$
bins. Low/high $k$ refers to $k_{\rm eff}=0.055$, 0.125 respectively;
low/mid/high $a$ refers to the ranges [0.37,0.5], [0.5,0.667], [0.667,0.833].
The $G_{\rm bin}-1$ column gives the fiducial value (i.e. the optimized values
from Sec. III) and the last two columns shows the $1\sigma$ uncertainties in
determining those values from the projected redshift space distortion data
($\sigma$ column), and also including a peculiar velocity measurement
($\sigma_{+}$ column; see Sec. IV.3).

The power to discriminate modifications from GR becomes greater when assessing
the joint, rather than 1D, likelihood. Figure 5 shows the marginalized 2D
joint confidence contours for all combinations of the $G_{\rm bin}$. Note that
the parameters are fairly independent, with the greatest correlation
coefficient being $-0.6$ between $G_{21}$ and $\Omega_{m}$; between the
$G_{\rm bin}$ themselves, the maximum amplitude correlation is $-0.4$.

Figure 5: Corner plot showing 2D joint 68% CL constraints on the gravitational
strength parameters in the two bins of $k$ and three bins of redshift. The red
$\times$ denotes the GR value (0,0). The solid black contours use redshift
space distortion data over $z=[0.2,1.7]$; the blue dashed contours add
peculiar velocity data at $z=0.1$ (see Sec. IV.3). The matter density
$\Omega_{m}$ is marginalized over with a prior of 0.01.

The red $\times$’s show the GR value in each panel, and we see that at low $k$
there is no statistically visible deviation from GR, even among the
crosscorrelations (upper tip of the triangle). At high $k$ we see GR lies at
or outside the 68% CL contour, and the deviation is especially noticeable in
the crosscorrelations (right tip of the triangle). The overall difference in a
GR fit to growth rate data from a cosmology with the fiducial modified gravity
is $\Delta\chi^{2}=+20$, with 6 fewer parameters, giving at least a hint that
one should look beyond GR. As in Sec. IV.1 we find that tightening the prior
on $\Omega_{m}$ gives little improvement since there is not much covariance.
Only a further generation of more precise data could improve the constraints
on gravity in this model independent approach – or combination with other
probes of growth.

###  IV.3 Impact of Low Redshift Data

Another method being implemented with current surveys, such as DESI, to
measure directly the growth rate is peculiar velocities [36, 37]. This can
improve the precision at low redshift dramatically over redshift space
distortions, which are limited by low available volume for clustering
statistics. We consider the addition of a single data point for $f\sigma_{8}$
at $z=0.1$ with precision 3% at each of $k_{\rm eff}=0.055$, 0.125 and
investigate its impact on the gravitational constraints. This precision is a
rough approximation of what DESI spectroscopy may provide from peculiar
velocity measurements [36, 37].

This single redshift data point improves the constraints on the binned $G_{m}$
approach as shown by the last column in Table 2 for the 1D uncertainties and
the dashed blue contours in Fig. 5 for the 2D joint constraints. Leverage is
added particularly in the late time bin, i.e $G^{\rm bin}_{i3}$, such that GR
is now pulled to the edge of many of the 2D joint confidence contours. The
greatest effect is on $G_{13}$–$G_{23}$, where the uncertainty area is reduced
by a factor 5.8. The total $\Delta\chi^{2}$ distinction from GR is now
$\Delta\chi^{2}=28$, i.e. the one peculiar velocity data point distinguishes
from GR by $\Delta\chi^{2}=8$.

For the scalar-tensor Padé approach we also find that the peculiar velocity
data on the growth rate at $z=0.1$ can have good complementarity with the
redshift space distortion measurements. While this one added data point does
help to determine the amplitude parameter $c$, lowering $\sigma(c)$ from 1.7
without the $z=0.1$ data to 0.8 with, this is still not sufficient. Fixing
$c=4/3$ as for $f(R)$ theories as discussed in Sec. IV.1, the peculiar
velocity data has excellent leverage, as seen in Fig. 6. Now
$\sigma(M_{a})=0.014$, down from 0.020, and $\sigma(M_{\rm dS})=0.094$, down
from 0.181. The overall uncertainty area reduces by a factor 2 by adding the
$z=0.1$ peculiar velocity measurement. Using the same naive procedure as
before to estimate $f_{R0}$, the $1\sigma$ upper limit on $M_{a}+M_{\rm dS}$
is 0.19, for a lower limit on the fiducial model of $|f_{R0}|\gtrsim 2.9\times
10^{-6}$.

Figure 6:  Joint 68% confidence contours for $M_{a}$ and $M_{\rm dS}$ with
fixed $c=4/3$ and a prior of 0.01 on $\Omega_{m}$, for redshift space
distortion only (solid curve, as in Fig. 4), and with the peculiar velocity
data at $z=0.1$ (dashed blue curve).

##  V $2\times 2$ Bins

To get clearer distinction from GR we could use fewer bins in $a$, albeit at
the loss of some evolutionary information on $G_{m}$. To explore this, we
consider two bins in $a$ for $G_{m}$: $a=[0.25,0.556]$ and $a=[0.556,0.833]$,
with the dividing line at $z=0.8$ so there will be sufficient precise data in
both bins. The two bins in $k$ remain as is, with a piecewise constant form
for the $G_{m}$ bin values. For just two bins in $a$, piecewise constant
$G_{m}$ will not give a sufficiently accurate $\widehat{f\sigma_{8}}$ relative
to the exact $f\sigma_{8}$, so we use a piecewise linear form, with continuity
at the bin boundary $z=0.8$. We call our new $2\times 2$ gravity parameter set
$G_{1A}$, $G_{1B}$, $G_{2A}$, $G_{2B}$, where the first index indicates
low/high $k$ as before, and the second index $A/B$ indicates low/high $a$. (We
change from numbers to letters to avoid confusion with the $G_{11}$ etc. from
the $2\times 3$ set.) The bin values are defined at the midpoint in $a$ of
each bin.

Figure 7 illustrates that the maximum deviation in reconstructing
$f\sigma_{8}$ can be under 0.2% even with only two bins in $a$. As before we
use bin centers at $k_{\rm eff}=0.055$, 0.125 and optimize the values of
$\delta G^{\rm bin}$ to minimize the rms fractional deviation of
$\widehat{f\sigma_{8}}$ from $f\sigma_{8}$ given by the fiducial scalar-tensor
model. The optimized values are $\delta G_{1A}=0.011$, $\delta G_{1B}=0.048$,
$\delta G_{2A}=0.049$, $\delta G_{2B}=0.15$.

Figure 7: The fractional difference of $f\sigma_{8}$ using the optimized
$2\times 2$ binned $G_{m}^{\rm bin}$ (giving $\widehat{f\sigma_{8}}$) vs using
the exact modified gravity $G_{m}$ (giving $f\sigma_{8}^{\rm Exact}$) is
plotted vs $a$ for various $k$. As in Fig. 3, only the values at $k_{\rm
eff}=0.055$, 0.125 are observationally relevant. Similarly, the relevant (i.e.
good data precision) range in $a$ is indicated by the unshaded region.  Figure
8: Corner plot showing 2D joint 68% CL constraints on the gravitational
strength parameters in the two bins of $k$ and two bins of redshift. The red
$\times$ denotes the GR value (0,0). The solid black contours use redshift
space distortion data over $z=[0.2,1.7]$; the dashed blue contours add
peculiar velocity data at $z=0.1$ (see Sec. IV.3). The matter density
$\Omega_{m}$ is marginalized over with a prior of 0.01.

Therefore we can use the $2\times 2$ bin parametrization for robust comparison
to projected data, and derive constraints on the gravity parameters. The
corner plot of the 2D joint confidence contours is shown in Fig. 8. Now,
several panels show GR as lying significantly outside the 68% confidence
contour, either using redshift space distortion data alone or adding the
peculiar velocity data at $z=0.1$. In some cases it lies outside the 95% joint
confidence limit.

As before, the discrimination from GR is most statistically significant for
the higher $k$ bin. The higher redshift bin also has tighter constraints than
the lower redshift bin. The peculiar velocity data continues to exhibit strong
leverage in complementarity with redshift space distortion measurements. In
the $2\times 2$ case, the total distinction from GR is now
$\Delta\chi^{2}=23$, for two fewer parameters than the $2\times 3$ case. This
improves to $\Delta\chi^{2}=32$ with the peculiar velocity data point at
$z=0.1$. The greatest improvement due to adding peculiar velocity data is on
the $\delta G_{1B}-\delta G_{2B}$ joint contour, where the overall uncertainty
area is reduced by a factor of 7.

##  VI Conclusions

The growth rate of cosmic structure is now being measured precisely by the
current generation of spectroscopic surveys. This can combine with expansion
history information to tighten constraints on dark energy and cosmic
acceleration. However, beyond general relativity the growth rate carries extra
information on the theory of gravitation, and for many theories this includes
scale dependence even in the linear to quasilinear density regime.

We have demonstrated that the scale dependence in canonical scalar-tensor
theories of gravity can be accurately accounted for by a model independent
binning of the gravitational coupling: 2 bins in $k$ and 3 bins in $a$ deliver
0.02-0.27% rms accuracy in the full growth rate $f\sigma_{8}(k,a)$ even for a
fairly conservative scalar-tensor gravity theory for comparison to
observations. Such model independent bins have little covariance and are
readily interpretable.

If one does not use all the physics present in scalar-tensor gravity, e.g.
uses a power law in $a$ for $G_{m}$ rather than the true Padé form, or
neglects the freezing of the scalaron mass as the universe is dominated by
cosmic acceleration, we show that the results will be strongly different, both
quantitatively and in interpretation. Some other models of gravity may contain
a constant offset, i.e. constant deviation from GR even at low $k$, and these
tend to be easier to constrain but address different physics.

Fitting parameters in the scalar-tensor form itself, thus being somewhat more
model dependent, does not work due to strong covariances. Only when a
particular class is chosen, such as $f(R)$ gravity, to fix the amplitude
parameter $c$ can the shape parameters $M_{a}$, $M_{\rm dS}$ describing the
scale dependence be fit with reasonable precision. These can be related to
more specific quantities such as $|f_{R0}|$ and we demonstrate potential
distinction from general relativity by obtaining a lower bound
$|f_{R0}|\gtrsim 1.4\times 10^{-6}$ (in contrast to a spuriously tight
$[0.8,1.2]\times 10^{-5}$ when ignoring the necessary de Sitter physics) for a
model with true $|f_{R0}|\approx 10^{-5}$.

Of great promise is the strong complementarity of low redshift peculiar
velocity probes of the growth rate with redshift space distortions from
surveys at higher redshift. Recall that the growth rate at low redshift is
sensitive to the cosmic gravitational history at all higher redshifts.
Peculiar velocity data at $z\approx 0.1$ can tighten the 2D joint uncertainty
area on binned gravity parameters by up to a factor 7, and raise the
statistical significance of discrimination from general relativity to several
measurements of 95% confidence level, or an overall $\Delta\chi^{2}=28$.

Finally, we also explore an alternate model independent parametrization with
$2\times 2$ bins of $G_{m}$ and find it also works well. It accurately
reproduces the full scalar-tensor prediction for $f\sigma_{8}(k_{\rm eff},a)$
with two fewer parameters. The constraints from projected data tighten, with
greater discrimination power from GR. Two downsides are a more coarse grained
view of the evolution in $a$, and a somewhat greater correlation between bins
(because of the needed continuous piecewise linear rather than piecewise
constant parametrization). One could envision using $2\times 2$ as an alert
for a deviation from GR, and then reanalyzing the data with $2\times 3$ to
pick up finer detail.

###### Acknowledgements.

This work was supported in part by the Energetic Cosmos Laboratory. EL is
supported in part by the U.S. Department of Energy, Office of Science, Office
of High Energy Physics, under contract no. DE-AC02-05CH11231.

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Cometary Ionospheres: An Updated Tutorial

Arnaud Beth
Umeå Universitet

Marina Galand
Imperial College London

Cyril Simon Wedlund
Österreichische Akademie der Wissenschaften, Institut für Weltraumforschung

Anders Eriksson
Institutet för Rymdfysik

= 11pt
This chapter aims at providing the tools and knowledge to understand and model the plasma environment surrounding comets in the innermost part near the nucleus. In particular, our goal is to give an updated post-Rosetta `view' of this ionised environment: what we knew, what we confirmed, what we overturned, and what we still do not understand.

§ INTRODUCTION

§ BIRTH OF A COMETARY IONOSPHERE

§.§ Ion continuity equations

§.§ Source of the cometary plasma

§.§.§ Photoionisation

§.§.§ Electron-impact ionisation

§.§.§ Charge-exchange and ionisation by solar wind ions

§.§.§ Which ionisation sources matter at comets?

§.§ Centre of chemical reactions

§.§ The Ionosphere: Putting it all together

§.§.§ Total ionospheric density

§.§.§ A rich diversity of ions: a real ion zoo

§ ELECTRON POPULATION AND ITS IMPACT

§.§ Different electron populations

§.§.§ Newly-born warm, cometary electrons

§.§.§ From warm to cold cometary electrons

§.§.§ Accelerated electrons

§.§ Effect on the ion dynamics

§.§ Diamagnetic cavity

§ ION POPULATION

§.§ Collisions versus acceleration

§.§ Pick-up ions

§.§ Solar wind versus ionospheric plasma

§ DUSTY (DIRTY) COMPLEX PLASMA

§.§ Why care about dust charging?

§.§ Dust charging

§.§ Charged dust dynamics at Comet 67P

§ SUMMARY & OPEN QUESTIONS

We would like to warmly thank P. Stephenson (Imperial College London, UK) and G. Wattieaux (Université of Toulouse, France) for their help and support, in particular in providing un/processed dataset shown in Fig. <ref> and in generating Fig. <ref>, respectively. We warmly thank Z. Lewis for her careful reading of the chapter and her suggestions.
A.B. and his work at Umeå University was supported by the Swedish National Space Agency (SNSA) grant 108/18. C.S.W. thanks the Austrian Science Fund (FWF) P32035-N36. Work by M.G. at Imperial College London was supported in part by STFC of UK under grant ST/W001071/1.