For fossil mammals, the number of phylogenies is increasing, cla- distic and otherwise (Fraser et al., 2015), and in the absence of phylogenetic information, taxonomy can also be used (Soul & Friedman, 2015).

There are numerous phylogenetic comparative methods that can be put to good use by scientists using tooth wear including, but not limited to, phylogenetically independent contrasts, phylogenetic generalized least squares, phylogenetic principal components analysis, and PDFA.

All of these analyses are now available in the free software platform R in packages such as ape, geiger, and caper (Harmon et al., 2008; Orme et al., 2011; Paradis, 2006).


• Mammal diets and tooth wear show significant phylogenetic sig- nal when using Pagel's λ and phylogenetic heritability (H 2 ).


• We suggest that phylogenetic conservation of functional traits re- lated to feeding explains phylogenetic signal in mammal diet and tooth wear.


• The ways in which teeth wear are determined, in part, by di- etary constraints placed on animals by their morphology and the influence of morphology on the ways in which the teeth occlude.


• Tooth wear is not merely phylogeny in disguise; it can vary among individuals and time periods, but the degree of variation is con- strained by morphology and thus phylogeny.


• Application of common, well-documented phylogenetic compara- tive methods can circumvent statistical issues related to noninde- pendence among tooth wear data points due to PNC.


• Restriction of tooth wear comparisons to closely related species with similar dental morphologies will reduce the need for phylo- genetic comparative methods but limit within-guild comparison of distantly related taxa.


Dear Editor, A new study by Fraser, Haupt, and Barr (2018) urges the use of phylogenetic comparative methods, whenever possible, in analyses of mammalian tooth wear.

We are concerned about this for two reasons.

First, this recommendation may mislead the research community into thinking that phylogenetic signal is an artifact of some sort rather than a fundamental outcome of the evolutionary process.

Secondly, this recommendation may set a precedent for editors and reviewers to enforce phylogenetic adjustment where it may unnecessarily weaken or even directionally alter the results, shifting the emphasis of analysis from common patterns manifested by large clades to rare cases.


Fraser et al. ( 2018) take phylogenetic niche conservatism as a starting point for their study, implying that species are constrained by ancestral traits and therefore have a “reduced dietary ability over evolutionary time.” That is plausible, as per our hypothetical example above.

However, what is misleading is the follow-up statement that “dietary inferences from tooth wear are biased by phylogenetic relatedness,” inferred from the fact that “tooth wear dietary proxies show strong phylogenetic signal” [1].

Methods of determining diet from tooth wear, as well as any other analytical methods of determining diet such as stable isotopes or fecal analyses, do not assume or require diet to be independent of phylogenetic affinity.

What they assume is that within the domain of validity of a method, each diet leaves a detectable trace (i.e. it wears teeth in a distinct way or leaves a distinct composition of isotopes), no matter which taxon the tooth comes from.


This gives these proxies the flexibility to not only monitor similarities in diet when they might be expected (e.g. between certain distantly related taxa), but to document differences in diet when they might not necessarily be expected based on tooth morphology alone.

Differences between closely related taxa have been captured, for instance, in bovids [2], cervids [3], ungulates [4], feliforms [5], canids [6], primates [7], and macropodids [8].

Indeed, many bioarchaeological studies have demonstrated distinctive and predictable diet-related differences in both gross dental wear and microwear within a single species, Homo sapiens [9].

The Equus and Bos example, pointed out by Fraser et al. ( 2018) as a case of possible problem with phylogenetic non-independence, in fact shows the opposite: Distantly related taxa, with convergently similar diet, have very similar results from all dental wear proxies [10].

Further, experimental studies have demonstrated different microwear attribute values in the same species when fed foods with different textural properties and/or grit loads [11].

In many cases, the diets of extinct taxa are also vastly different from those of extant taxa, despite being members of the same family [12].

Function-driven methods, such as mesowear or microwear, therefore give us the chance to recognize both similarities and differences in paleobiology that might not be detectable via traditional morphological or taxic techniques.


The same sort of thing can be said for stable isotope techniques where, for instance, experimental manipulations of diet result in changes to the isotopic composition of sampled tissues [13] and where, for example, carbon isotope values for bovids from the early hominin sites in South Africa indicate that 25% of specimens would be misclassified (in terms of diet) if based on taxonomic affiliation alone [14].

The fact that diet is not independent of phylogeny is one of the fundamental outcomes of the evolutionary process, since a descendant species is never created de novo but builds upon inheritance from its immediate ancestor.

“ Species of the same genus have usually, though by no means invariably, some similarity in habits and constitution, and always in structure” [15], this is a phylogenetic signal.

Phylogenetic niche conservatism, however, results when closely related species are more similar ecologically than would be expected based on their phylogenetic relationships [16].

Therefore, the phylogenetic signal itself, as tested by Fraser et al. ( 2018), would not indicate evolutionary constraints; however, its residual as compared to random divergence could indicate it [17].

Multiple studies (reviewed by Losos, 2008) have found that phylogenetic niche conservatism is by no means omnipresent, while phylogenetic signal is the norm.


And even if lineages remain conserved within their diets and habitats, it does not follow that functional relationships between (dental) traits and diets become decoupled.

If teeth of the same kind remain useful for the same diets, like carnassials for eating meat, inference of diets via dental proxies would not be biased.

An analogy can be made to an X-ray machine used at airports to detect the presence of sharp objects.

Suppose a family arrives that have jointly prepared for a picnic, and therefore, all have knives in their pockets.

If the machine correctly detects all the knives, it would yield the equivalent of a phylogenetic signal in its outputs.

This does not mean that the method by which the machine detects knives is biased, or needs to be adjusted. And if we adjust the machine in order to decrease phylogenetic signal, in this situation, the result will be undoubtedly worse; the machine will either make false alarms or miss real detections (Type I and Type II errors).

Tooth wear is a function of behaviors occurring in life.

It is fundamentally different from genetically mediated dietary proxies, such as occlusal morphology, so its analysis should not be constrained by the use of phylogenetic comparative methods.


We do not mean to imply that phylogenetic methods are wrong as such. They have their meaningful uses but their relevance depends on the research question rather than the source of the data.

Yet a statement received by some of us from a recent reviewer echoes a worrying trend to enforce phylogenetic adjustments everywhere where data of phylogenetic origin are used: “Although I agree with the authors’ rationale for not wanting to use PGLS1, in this day and age it is simply not acceptable to NOT do the analyses using PGLS.

” We argued that this was unnecessary, and the editors were in agreement.

Analysis of phylogenetically related data is not trivial, and there is no solution that fits all purposes (see Westoby, Leishman, and Lord (1995), Losos (2008), Revell (2010) for comprehensive discussions).

We strongly disagree with the recommendation of Fraser et al (2018) to use phylogenetic comparative methods in studies of mammalian tooth wear whenever possible.

Phylogenetic regression is not by itself a better or more advanced analysis method than ordinary regression.

The choice of the methods and interpretations of results depends on the authors of analyses and the research questions they tackle.


In our recent study, we describe strong phylogenetic signal in tooth wear dietary proxies.

We believe that phylogenetic signal is detectable in tooth wear because the feeding apparatus acts as an evolutionary and functional module [1].

Due to the significant functional role of the feeding apparatus, changes in one component impact the other components of the module [2].

The result is a difference in chewing stroke, the ways in which the teeth contact each other, and, thus, the ways which the teeth wear.

We further argue that such modularity of the chewing apparatus results in certain functional and thus evolutionary constraints, potentially leading to phylogenetic niche conservatism (PNC) [3].

The end result is that tooth wear (like diet) shows strong phylogenetic signal.

In fact, our finding seems unsurprising given that much of the methodological practice in the application of analytical tooth wear methods acknowledges this phylogenetic signal [4].

We subsequently make a recommendation that, when possible and necessary, appropriate phylogenetic comparative methods (PCMs) be applied, a recommendation that has been made time and time again in the ecology and evolution literature [5].


DeSantis et al. ( 2018), however, express concern that (a) our study will cause editors to unnecessarily enforce the use of phylogenetic comparative methods during review and (b) that our study will mislead researchers into thinking phylogenetic signal is merely an artifact, rather than a result of the evolutionary process.

In our recent study, we do not suggest that tests for phylogenetic signal and PCMs be applied when they are unnecessary.

For example, studies focused on a small number of species within the same genus are unlikely to require such statistical corrections [6].

We also recommend authors be familiar with the limitations of tests for phylogenetic signal when their sample of species is small [7].

We strongly assert that the source of phylogenetic signal in diet and thus tooth wear is the evolutionary process of descent with modification from a common ancestor [8].

However, there is a complex relationship among evolutionary process, rate, and phylogenetic signal [9].

The implication of the DeSantis et al. ’s argument, however, is that if phylogenetic signal results from the evolutionary process, there is no need to correct for it statistically.

We disagree strongly with this implication.

Even if the phylogenetic signal in dietary proxies such as mesowear and microwear is "meaningful" in the sense that it is related to diet, this does not change the statistical difficulties introduced by phylogenetic nonindependence [10].

Furthermore, in no way does the presence of phylogenetic signal, nor suggesting that PCMs be used when necessary, imply that dietary proxies are unreliable indicators of diet.


Suppose we are using hypsodonty (presence of high crowned teeth) as a metric for inferring grazing behavior among mammals [11].

If we sample a number of mammals and use a discriminant function analysis to estimate the strength of association between hypsodonty and feeding behavior, we would find that hypsodonty is a very good indicator of a grazing lifestyle.

However, the gelada baboon, an open-area feeder whose diet is dominated by grasses, would be misclassified by the discriminant function, based on the absence of hypsodont cheek teeth [12].

Thus, the strength of the association between hypsodonty and grazing is over-estimated in this scenario.

Hypsodonty is an honest indicator of grazing, but it is constrained to certain phylogenetic groups, and there exist other ways to be a grazer that do not require hypsodonty.

In a further hypothetical example used by DeSantis et al. ( 2018), families on their way to the airport tend to be either "knife carrying" or "knifeless." We extend this example to suggest that, if the material from which the knives are made shows "phylogenetic signal," airport security might overestimate the strength of the link between beeping magnetometer alarm and the presence of sharp cutting tools.

While the beeping alarm is an honest signal of the presence of a knife, there are other equivalent knife-wielders that are not detectable by this method.

In our recent study, we use a phylogenetic discriminant function analysis to demonstrate the result of such overconfidence arising from phylogenetic signal on our ability to correctly classify mammals by diet using tooth wear [13].


DeSantis et al. ( 2018) further suggest that (a) convergence of diets among distantly related mammals and (b) intraspecific dietary differences negate the need to correct for phylogenetic signal in tooth wear.

We consider the following example.

Hypsodonty has evolved numerous times within Mammalia but; for our purposes, let us consider only artiodactyls and perissodactyls.

All extant equids are hypsodont.

Some clades of artiodactyls possess similarly high-crowned teeth such as members of the tribe Bovini and family Antilocapridae.

The presence of high-crowned teeth is notably nonrandomly distributed throughout Artiodactyla and Perissodactyla [14].

Hypsodont members of these clades have inherited their high-crowned teeth from a common ancestor.

In fact, as far as we are aware, hypsodonty never reverses; hypsodont taxa never return to a low-crowned condition.

So, we would attain significant estimates of phylogenetic signal for hypsodonty even though it has arisen multiple times via different developmental and genetic mechanisms.

Convergent evolution therefore does not negate the need for phylogenetic comparative methods.


Dietary change through time might indicate weak or absent PNC.

However, as acknowledged by DeSantis et al. ( 2018), PNC, and phylogenetic signal are not the same phenomenon [15].

Significant phylogenetic signal can result both from a Brownian Motion model of evolution, wherein a trait evolves via a random walk, as well as PNC, a phenomenon wherein closely related species are more similar than is expected given a Brownian Motion model of trait evolution [16].

Therefore, significant phylogenetic signal can occur in the absence of PNC.

The fact that intraspecific shifts in diet occur among mammals therefore has no bearing on whether dietary phylogenetic signal exists within Mammalia or whether it needs to be corrected for.

The tooth wear of members of a species can and do vary through time but we suggest that they vary within some range that is determined by functional constraints of the feeding apparatus [17].

This manifests as dietary shifts, say from a less gritty diet to a more gritty diet, but should rarely manifest as an intraspecific shift from herbivory to carnivory, for example [18].

We further agree with DeSantis et al. ( 2018) that experimental diets can and do change the properties of mammal tooth wear [19].

However, our study refers only to the natural diet of mammals.

Feeding a lion rocks would most assuredly change how the teeth wear but has no bearing on whether the natural diets of mammals show phylogenetic signal and indicates nothing about how common PNC is among mammals.


Finally, DeSantis et al. ( 2018) rightly indicate that most PCMs assume that the residuals of y ~ x from an ordinary least squares regression are statistically independent [20].

In cases where the x and y variables show no autocorrelation but the residuals of y do, ordinary least squares regression performs poorly [21].

Herein, we demonstrate that when both x and y show high phylogenetic signal, the λ estimates for the residuals of y consistently indicate high phylogenetic signal [22].

Therefore, justified in recommending the use of PCMs.

Furthermore, Revell (2010) recommends fitting a phylogenetic generalized least squares model using maximum likelihood to simultaneously estimate λ and the regression coefficients.

If λ is zero, PGLS converges on the ordinary least squares solution.

Therefore, the application of a PCM, specifically PGLS, from the outset should not mislead researchers.


In conclusion, we agree with DeSantis et al. ( 2018) that phylogenetic comparative methods need not be applied when their use is unnecessary or unjustified such as during a study of intraspecific dietary variation.

However, we note that application of phylogenetic comparative methods (most notably phylogenetic generalized least squares) when phylogenetic signal is low will not lead researchers astray [23].

Our recent study thoroughly describes the fact that phylogenetic signal can arise from descent with modification and PNC.

We thus also agree with DeSantis et al. ( 2018) that phylogenetic signal in the context of tooth wear not be viewed as a mere inconvenient artifact. But we disagree with their implication that PCMs are therefore unnecessary.

If authors are interested in interpreting p-values from comparisons of species from disparate clades, they should have an understanding of the structure of their data as it pertains to phylogenetic signal.


Based on all the findings about the possible treatment in the pre- clinical phase of RA, reliable biomarkers are needed to identify patients who are on the trajectory to develop RA.

Once a panel of biomarkers has been found and carefully validated, popula- tion screening for evidence-based, effective treatment is a con- ceivable approach in the efforts to prevent RA.

In this context, recently, the combination of ACPAs, RF and anticarbamylated peptide (CarP) autoantibodies 5 has been shown to provide a very high odds ratio for RA.

6 Unfortunately, RF and anti-CarP autoantibody data were not available for the population described by van Zanten et al. 1


As the Lifelines Study is a prospective longitudinal cohort study with a targeted 30-year follow-up, it might be possible to gain follow-up information on this study population.

It will be interesting to see which partici- pants will eventually develop RA.

In addition, it would be valu- able to test the stored serum samples for as many biomarkers as possible in order to identify patients who develop RA in a certain period of time.

Lastly, combining biomarker data with clinical parameters might result in sufficient power to precisely predict the development of RA.

7 Several studies have now repeatedly shown that ACPAs and other biomarkers (eg, autoantibodies, inflammatory proteins, cytokines and microRNA) can antedate the development of RA by many years.

7 8 Although these data are intriguing, it would be more valuable to have biomarkers that are able to provide insights into the evolution of RA within the next 6-12 months, corresponding to the so-called window of opportunity.

9 10 When it comes to population screening, health economics becomes an important factor in the equation.

Since the global healthcare expenditures are constantly increasing and reaching non-sustainable thresholds in many jurisdictions, health eco- nomic studies of direct and indirect costs will be needed to dem- onstrate that investments in screening for pre-RA and early interventions or therapies are not outweighed by poor clinical outcomes.

Based on remarkable healthcare expenditures on the management of RA, there is a significant opportunity to achieve meaningful savings.

2 11 12


The design of the different RA prevention trials is rather diverse in terms of the treatment used and the inclusion criteria for the individuals at risk to be treated.

2 Therapeutic agents range from hydroxychloroquine to biologics (eg, tumour necrosis factor- alpha inhibitors).

Different treatments surely require also differ- ent timings in the preclinical phase of the disease.

Lastly, infor- mation as to who will develop RA in the distant future might be more useful in case preventive drugs (such as vaccines) become available.


Anticitrullinated protein/peptide antibodies (ACPAs), in addition to rheumatoid factor (RF), represent a serological hallmark in the diagnosis of rheumatoid arthritis (RA).

In this context and with interest I read the recent article by van Zanten et al [1] summarising the results from the Dutch ‘Lifelines Study’, a large population-based study of 40,132 individuals.

In this setting and using an adjusted cut-off for the ACPA assay, the prevalence of ACPAs was 1.0% and associated older age, female gender, smoking, joint complaints, RA and first-degree relatives with rheumatism.


The early identification of patients in the preclinical phase of RA is of high importance as it became evident during the last decade that early intervention can prevent joint damage in patients with RA.

Consequently, several ongoing studies are focused on the prevention of RA based on the treatment of individuals at high risk to develop RA [2].

All these prevention trials leverage the concept of the ‘window of opportunity’ to prevent or delay the clinical ravages and attending healthcare expenditures associated with RA.

Although a recent trial was unable to prevent the onset of RA (PRAIRI, unpublished data), the study clearly demonstrated that the treatment in the preclinical phase of RA postponed the development of the disease.

Most studies aim to treat patients as early as possible, an approach which seems intuitive.

However, it is unclear if treatment too early in the clinical course also leads to failure of response.

Most conventional approaches to the prevention of RA are based on the concept of restoring the lost tolerance of the immune system.


PRECLINICAL RA
The risk for the development of RA depends on many factors, which can be divided into two main categories: the modifiable (eg, smoking, behaviour) and the constant risk factors (eg, genetics) [3][4].

The preclinical phase of RA may be initiated by modification of the risk profile (eg, smoking) and is characterised by break of tolerance as part of the autoimmune processes, leading to increasing joint inflammation and eventually tissue damage and significant morbidity.

The development of RA has primarily been linked to factors that affect the gastrointestinal, the respiratory and reticuloendothelial systems [4].

An additional challenge to identify the right time for treatment is the patient-specific rate of the evolution from preclinical to clinical RA.

In some patients, the preclinical phase can take several years, whereas in other patients, this conversion happens in a shorter period of time.

Factors contributing, to defining or accelerating this transition are not fully understood.


BIOMARKERS
Based on all the findings about the possible treatment in the preclinical phase of RA, reliable biomarkers are needed to identify patients who are on the trajectory to develop RA.

Once a panel of biomarkers has been found and carefully validated, population screening for evidence-based, effective treatment is a conceivable approach in the efforts to prevent RA.

In this context, recently, the combination of ACPAs, RF and anticarbamylated peptide (CarP) autoantibodies [5] has been shown to provide a very high odds ratio for RA [6].

Unfortunately, RF and anti-CarP autoantibody data were not available for the population described by van Zanten et al [1].

As the Lifelines Study is a prospective longitudinal cohort study with a targeted 30-year follow-up, it might be possible to gain follow-up information on this study population.

It will be interesting to see which participants will eventually develop RA.

In addition, it would be valuable to test the stored serum samples for as many biomarkers as possible in order to identify patients who develop RA in a certain period of time.

Lastly, combining biomarker data with clinical parameters might result in sufficient power to precisely predict the development of RA [7].


Several studies have now repeatedly shown that ACPAs and other biomarkers (eg, autoantibodies, inflammatory proteins, cytokines and microRNA) can antedate the development of RA by many years [7][8].

Although these data are intriguing, it would be more valuable to have biomarkers that are able to provide insights into the evolution of RA within the next 6–12 months, corresponding to the so-called window of opportunity [9][10]. When it comes to population screening, health economics becomes an important factor in the equation.

Since the global healthcare expenditures are constantly increasing and reaching non-sustainable thresholds in many jurisdictions, health economic studies of direct and indirect costs will be needed to demonstrate that investments in screening for pre-RA and early interventions or therapies are not outweighed by poor clinical outcomes.

Based on remarkable healthcare expenditures on the management of RA, there is a significant opportunity to achieve meaningful savings [2][11][12].


CLINICAL PREVENTION TRIALS
The design of the different RA prevention trials is rather diverse in terms of the treatment used and the inclusion criteria for the individuals at risk to be treated [2].

Therapeutic agents range from hydroxychloroquine to biologics (eg, tumour necrosis factor-alpha inhibitors).

Different treatments surely require also different timings in the preclinical phase of the disease.

Lastly, information as to who will develop RA in the distant future might be more useful in case preventive drugs (such as vaccines) become available.


We would like to thank Michael Mahler for his letter, ‘Population-based screening for anticitrullinated protein antibodies (ACPA): A step in the pathway to the prevention of rheumatoid arthritis?’ In his letter, Dr Mahler makes a plea to analyse and define reliable biomarkers to identify subjects who are on the trajectory to develop rheumatoid arthritis (RA).

We agree with this notion of Dr Mahler as current efforts are increasingly focusing on the possibility to install early preclinical treatment to prevent progression to RA.


For example, several randomised placebo-controlled trials (RCTs) have been initiated recently.

These clinical trials will investigate the therapeutic potential of agents such as rituximab (PRAIRI study: NTR No. 1969), abatacept (APIPPRA study: ISRCTN No. 46017566 and ARIAA study: EudraCT No. 2014-000555-93), hydroxychloroquine (StopRA trial: NCT No. 02603146) and methotrexate (TREAT EARLIER: NTR No. 4853) in individuals at risk of RA.

A study evaluating the immunomodulatory effect of atorvastatin in patients with seropositive arthralgia has also been initiated (STAPRA study: NTR No. 22389).


Our publication contributes to these and other efforts as it describes the presence of a prominent biomarker for RA, ACPAs, in a large population-based study and thereby aids to the development of predictive algorithms identifying individuals at risk for RA development.

As outlined in this publication, we determined the prevalence of ACPA positivity and its association with known RA risk factors in 40,136 participants from the Lifelines cohort [2].

Lifelines is a large ongoing prospective population-based cohort study and biobank to investigate the interaction between environmental and genetic factors in the development of chronic diseases [3].

The standardised protocol includes physical examination, extensive questionnaires, and biobanking of serum, plasma, urine and DNA.


Participants will be followed up according to a fixed protocol for at least 30 years [4].

Within the existing infrastructure, additional collection of data and additional studies in already biobanked and future samples can be performed.

For example, the composition of the microbiome is currently being analysed in over 900 subjects participating in the Lifelines deep cohort [5], eventually facilitating the contribution of the microbiome composition to the risk to develop RA.

Likewise, genetics such as whole genome sequencing including human leukocyte antigen (sub-) typing can be performed, thereby allowing the analyses of the contribution of the microbiome in the susceptible genetic background to the development of autoimmunity [6].

Other described predictive markers like the acute-phase reactants erythrocyte sedimentation rate and C reactive protein and additional markers can be incorporated in the development of prediction models.


Indeed, recent insight into the specificity of the combined presence of ACPA, rheumatoid factor and anticarbamylated protein antibodies provides interesting possibilities in further narrowing down persons at risk [7][8].

Overall, Lifelines will give us the unique opportunity to gain follow-up information on our study population.

With funding, it will be possible to measure additional biomarkers as indicated above.

Therefore, this cohort is well suited for subsequent analyses on other biomarkers and risk factors, as also advocated by Dr Mahler.


Our future goal is to build a prediction model that can distinguish participants at high risk from those at low risk to develop RA.

Such a model is relevant because both animal data and clinical data suggest that prevention of RA may be possible and early treatment aiming at drug-free remission is possible [9][10].

The performance of RCTs on preventive treatment strategies in individuals at risk for RA is challenging.

The development of reliable assays and predictive algorithms to define a population with high enough at risk is crucial to allow the design of effective preventive therapy aiming to restore immune homeostasis with a great specificity, low toxicity and long-term effectiveness.


Underreporting of violent events has been defined as failure of victimized employees to report these events to their employers, the police, or other officials (Findorff, McGovern, Wall, & Gerberich, 2005).

Underreporting hinders violence prevention efforts in two ways.

First, underreporting results in an underestimation of the true extent of the problem, thus indicating less of a need for prevention of possible negative effects than may actually be warranted (Bensley et al., 1997).

Second, without knowledge of the full spectrum of violent events to which workers are exposed, prevention efforts can only be designed to affect limited aspects of the problem (Arnetz, 1998; Arnetz, Aranyos, Ager, & Upfal, 2011a).

In health care, various reasons for underreporting WPV have included lack of injury or time lost, time-consuming incident reporting procedures (Arnetz, 1998; Gates, 2004; Lanza & Campbell, 1991), lack of supervisory or coworker support, fear of reprisal or blame (Gates, 2004; Sato, Wakabayashi, Kiyoshi-Teo, & Fukahori, 2013), belief that reporting will not lead to any positive changes (Gates, 2004; Kvas & Seljak 2014), and the common perception among health care workers that violence is simply "part of the job" (Gates, 2004; Lanza & Campbell, 1991; Lanza, Schmidt, McMillan, Demaio, & Forester, 2011).

Varying definitions of violence among employees and within organizations (Arnetz, 1998; Sato et al., 2013) can also affect reporting behavior.


Physical assaults by patients, relatively common in emergency (Gacki-Smith et al., 2009; Gates, Ross, & McQueen, 2006; Taylor & Rew, 2011), psychiatric (Lanza et al., 2011), and geriatric (Leonard, Tinetti, Allore, & Drickamer, 2006; Zeller et al., 2009) departments, may not be reported if staff perceive that the aggressive behavior was unintentional, that is, related to the patient's illness (Hesketh et al., 2003; Lanza & Campbell, 1991; Sato et al., 2013).

Health care workers may also be reluctant to report non-physical violence from patients or co-workers because the behavior incurs no injury and may be organizationally sensitive, especially when it concerns the misuse of power by the perpetrator, such as when an employee is bullied by a supervisor (Ferns & Meerabeau, 2009; Findorff et al., 2005).


The magnitude of WPV underreporting among health care workers is difficult to quantify.

Questionnaires have been used to measure reporting behavior among workers (Ferns & Meerabeau, 2009; Findorff et al., 2005; Sato et al., 2013).

These studies have found a high prevalence of underreporting of both physical and non-physical violence, with all results based solely on employee self-report.

Earlier research compared different forms of incident reporting (Arnetz, 1998; Bensley et al., 1997; Grenade & Macdonald, 1995) and also found high rates of underreporting.

In a 12-month period, employees at 47 health care workplaces recorded 684 violent events on incident reporting forms developed for the research study; based on self-report, only 147 incidents (21%) were also filed as official work injury reports (Arnetz, 1998).


Bensley and colleagues (1997) compared assault injury rates among staff in a psychiatric hospital using compensation claims, hospital incident reports, and questionnaire data.

Rates differed widely: 13.8, 35, and 415 injuries per 100 employees per year were reported based on compensation claims, incident reports, and self-report, respectively.

That study used the hospital ward as the unit of analysis because questionnaires were anonymous and it was not possible to link individual responses to individual workers' compensation or work injury reports.

Grenade and Macdonald (1995) found student nurses' underreporting of physical assault using both documented incidents and a self-report questionnaire.

However, that study only compared results via the two methods without linking the two data sets.


To the researchers' knowledge, no study to date has linked individual questionnaire responses about WPV exposure and reporting behavior with actual incident documentation.

Using individual workers as the unit of analysis could quantify the magnitude of violence underreporting and identify specific worker characteristics associated with underreporting.


The current study compared individual questionnaire responses with actual documentation of WPV events in a cohort of hospital employees.

The aim of the study was to increase understanding of underreporting by investigating differences between self-report and actual documentation practices, and explore characteristics and reporting patterns of health care workers who underreport.

Based on previous research (Arnetz, 1998; Bensley et al., 1997; Grenade & Macdonald, 1995), it was hypothesized that reports of violence via questionnaire would exceed the number of actual documented incidents.

Furthermore, it was expected that reporting would be highest among hospital employees who were injured as a result of a WPV incident (Findorff et al., 2005; Sato et al., 2013) and among those working on psychiatric and emergency units (Arnetz, Hamblin, Ager, Aranyos, Upfal, et al., 2014).


The study was conducted in an American hospital system comprised of seven hospitals and approximately 15,000 employees.

The hospital system maintains a centralized electronic database of employee-reported occupational accidents and incidents, including needlestick injuries, slips, trips and falls, and violent events.

Incident reports are documented by employees via any hospital system computer.

The current study used a subset of the database that included only WPV data.

Hospital employees are encouraged to report all types of WPV, both physical and non-physical, including incidents perpetrated by patients or visitors, Workplace Health Saf.

Author manuscript; available in PMC 2016 August 31. known as Type II violence, and those perpetrated by other employees, Type III violence (Injury Prevention Research Center [IPRC], 2001).

The violence database was linked to the human resource database that provides information on employee age, gender, job category, date of hire, employment status, and paid productive hours (PPH).

This linkage enabled the calculation of standardized rates of violence per 100 full-time equivalents (FTEs) per year, thus providing the hospital system with comparison rates of violence occurrence across hospitals, work units, and over time.


This population-based surveillance and reporting system and WPV database have been described previously (Arnetz et al., 2011a; Arnetz, Aranyos, Ager, & Upfal, 2011b).

Hospital system policy required employees to document all WPV incidents, both with and without resulting injury, via the electronic reporting system or to a supervisor (Arnetz, Hamblin, Ager, Aranyos, Essenmacher, et al., 2014).

The current study was limited to employees on 42 hospital units (N = 2,010) across the hospital system.

Based on analysis of rates of WPV from a 30-month period (January 2010-June 2012), these units were identified as being at increased risk for violence (Arnetz, Hamblin, Ager, Aranyos, Upfal, et al., 2014).


The questionnaire developed for the study measured employees' experience with violence and aggression at work during the past year.

Socio-demographic/background items included age, gender, place (hospital) of employment, job category, supervisor status (yes/no), length of employment in the health care field, and length of employment within the hospital system.

Violence was defined as acts or threats of physical or verbal aggression.

Employees were asked whether they had been a target of violence or aggression at work during the past year.

Response alternatives were "No, never," "Yes, once or twice," and "Yes, several times" (Arnetz & Arnetz, 2001).

Violence-related injury: Employees were also asked whether they had sustained any physical injury as a result of a violent incident (No, none; Yes, mild injury; Yes, serious injury) and whether they had lost time from work as a result of a violent event (No, Yes).

Four items concerned reporting of violent incidents and asked whether employees were familiar with the centralized system for reporting incidents of WPV (No, Yes); whether they had reported a violent incident via the system during the past year (No, never; Yes, once or twice; Yes, several times); if employees had not reported a WPV incident in the electronic system, they were asked to provide a reason for not doing so; and whether they had reported WPV another way: to a supervisor, via the Compliance Hotline, a toll-free number that an employee may call to report any type of work-related issue anonymously, or by some other means.


Underreporting was defined as the percentage of employees who self-reported experiencing a WPV event but did not report any events into the electronic system.

Current hospital system policy mandates employees report any known incidents of violence through the electronic system or to a supervisor.

Supervisors must record all reported incidents through the electronic system within 24 hours from the end of the shift.

Thus, incidents reported by employees to their supervisors are theoretically entered into the system, either by the employee or the supervisor.


In the spring of 2013, employees assigned to all 42 hospital units (N = 2,010) were asked by the researchers to participate in a questionnaire survey regarding their exposure to WPV and knowledge of the WPV reporting system.

Questionnaires were mailed home to employees along with a postage-paid return envelope and a cover letter.

The letter described the purpose of the study and informed employees that participation was voluntary and questionnaire responses were anonymous.

Employees gave their consent to participate in the study by completing and returning the questionnaire.

Each employee responding to the questionnaire received a US$10 gift card by mail.

The cover letter explained that each questionnaire was coded with an identification number that enabled the research team to identify respondents from a master list.

Once the questionnaires had been returned and the gift cards had been mailed out, the list linking identification numbers with respondent names and addresses was destroyed.


Approximately 2 weeks after the first mailing, reminders and questionnaires were re-sent to non-respondents.

Approval for this study was granted by the Institutional Review Board at the university and the Research Review Council of the hospital system.


Using the pseudo identification numbers on the questionnaire survey, individual questionnaire responses were compared with actual events entered into the electronic reporting system in the previous 12 months.

Questionnaire data were linked to the database by a hospital system data analyst; the research team only had access to de-identified data from the hospital system database.


Chi-square analysis was used to compare questionnaire respondents with non-respondents in regard to gender, age group, employment status (full-time, part-time, and contingent), job category, job tenure, and documentation of WPV incidents via the electronic system.

Underreporters, that is, employees who reported WPV through this study but did not document any incidents in the electronic system, were compared with reporters, that is, employees who both self-reported an incident through this study and documented incidents in the system, using chi-square statistics.

In a final step, forward stepwise logistic regression was used to determine factors associated with reporting.

The dependent variable, reporting (yes/no), was calculated as having reported exposure to WPV on the questionnaire and also reported an incident in the electronic system.

An additional logistic regression was conducted to assess factors associated with reporting a violent event to a supervisor.

Independent variables in both regressions included employee age, gender, type of work unit, job tenure, employment status, violence-related injury, and lost work time due to a violent event.


A total of 446 employees responded to the questionnaire (response rate 22%).

More than 80% of respondents (n = 364) were female and 35% (n = 157) were 50 years of age or older.

The majority of respondents (n = 269, 60%) were nursing staff, part-time employees (n = 194, 44%) and worked in either acute care nursing units (n = 157, 35%) or emergency departments (n = 120, 27%).

Respondents differed significantly from non-respondents with Workplace Health Saf.

Author manuscript; available in PMC 2016 August 31.

regard to age (60.1% respondents vs. 49.1% non-respondents ≥ 40 years, p < .001), job tenure (17.5% respondents vs. 11.6% non-respondents worked ≥ 20 years in the hospital system, p < .01), and job category (5.2% patient care associates among respondents vs. 12.6% among non-respondents, p < .001).

Respondents did not differ significantly from non- respondents with regard to documentation of a WPV incident in the electronic system in the past year (8.3% respondents vs. 6.5% non-respondents, p = .19).


Figure 1 presents a flowchart summarizing survey responses and hospital system documentation of violent workplace incidents.

Of the 446 employees who responded to the questionnaire, 275 (62%) reported experiencing at least one WPV event in the previous year; of these, 243 did not document an incident in the database, representing an overall rate of underreporting of 88%.

Surprisingly, 63 of the 275 who self-reported violent events (23%) also reported having documented at least one incident via the electronic reporting system, but only 12 (4%) actually did so.

Among the remaining 212 employees who stated in the questionnaire that they did not document a violent incident electronically, 20 (9%) actually did so.

Thus, in reality, only 32 of the 275 employees (12%) who self-reported a violent event had formally documented the incident via the electronic reporting system.


Table 1 summarizes and compares characteristics of employees who documented violent events in the electronic reporting system ("reporters," n = 32) with employees who did not ("underreporters," n = 243).

No significant differences were found between the two groups with regard to gender, age, or length of employment.

Reporters included a significantly greater proportion of full-time employees than underreporters, and no contingent workers were found among reporters (p < .05).

Fifty percent of reporters (n = 16) had been injured as a result of a WPV incident, compared with 11.5% (n = 28) among underreporters (p < .001); 25% of reporters (n = 8) had lost time from work due to a WPV incident compared with 4.5% (n = 11) among underreporters (p < .001).


Of the 275 questionnaire respondents who said they had experienced a WPV incident, 45% (n = 125) reported the violence using an alternative method (Table 2).

Of the 32 who did report via the electronic system, 14 (43.8%) also reported in some other way.

Among the 243 underreporters, who did not report an incident via the electronic system, 111 (45.7%) did report using an alternative method.

Reporting to a supervisor was the most common alternative method; slightly less than 80% of both reporters and underreporters reported these incidents to their supervisors.


Table 3 summarizes the reasons for not reporting a WPV incident as described by the 212 questionnaire respondents who stated they had experienced a violent event, but had not reported it.

As indicated in Figure 1, 20 of the 212 individuals actually had documented an incident in the electronic system; thus, Table 3 compares results by reporters and underreporters.

The most common reasons for not reporting were that the individual had not been a target of/witness to a violent event (29.9%) and did not believe that reporting leads to change (28.4%).

The only significant difference between reporters and underreporters Workplace Health Saf.

Author manuscript; available in PMC 2016 August 31. concerned being unsure of how to report; surprisingly, this was more common among reporters (20%) than underreporters (5.4%) although only 14 respondents in total had selected this alternative.

Overall, 43 respondents had noted "other" reasons for not reporting including fear of retaliation (n = 13); the incident was reported by someone else (n = 6); the incident was patient-related (psychological or cognitive impairment) and was included in either patient notes or discussed among the health care team (n = 6).


Results of the logistic regression that examined factors associated with reporting incidents of violence in the electronic system are summarized in Table 4.

Employees had a higher likelihood of reporting via the electronic system if they incurred a physical injury as the result of the violent incident (OR = 6.22) or if an incident resulted in time away from work (OR = 3.56).

No other demographic or work-related factors were significant.


Factors associated with reporting WPV to a supervisor are summarized in Table 5.

Having worked less than 5 years (OR = 0.42) and working as security staff (OR = 0.22) were both significantly associated with lower likelihood of reporting WPV to a supervisor.


The aims of this study were (a) to compare the self-report of WPV via the questionnaire with actual documentation of violent incidents in a cohort of hospital employees, and (b) to examine the individual and job characteristics and reporting patterns of "reporters" versus "underreporters." As hypothesized, a greater proportion of questionnaire respondents (62%) self-reported an incident of WPV in the previous 12 months, compared with 12% who actually documented the incident via the electronic reporting system.

This finding is consistent with previous research (Ferns & Meerabeau, 2009; Findorff et al., 2005; Sato et al., 2013) based solely on self-report.


Findorff and colleagues (2005) conducted a study in a single health care organization and found that less than 60% reported physical violence and less than 50% reported non-physical violence to their employers; most reports were oral and not otherwise documented.

Ferns and Meerabeau (2009) reported that 45% of nursing students experienced verbal abuse during their clinical training and the majority (63%) stated they had reported the incident.

However, only four incidents were documented in writing.

Sato and colleagues (2013) found that more than 30% of nurses reported experiencing patient aggressive behavior in the previous month, but 70% did not report the incident.


The second hypothesis was that reporting would be highest among hospital employees who were injured, and among those working on psychiatric or emergency units; this was partially supported.

A significantly greater proportion of reporters had been injured or lost time from work as a result of a violent event, compared with underreporters; these two factors were also significantly associated with a higher likelihood of reporting through the electronic system.

This finding may in part be explained by hospital system policy, which states that employees who have been injured on the job and seek care at Occupational Health Services must document the incident in the electronic system.

These findings support previous research that found a higher likelihood of reporting when the symptoms or impact of the violence were more severe (Findorff et al., 2005; Sato et al., 2013).

Both studies were based Workplace Health Saf.

Author manuscript; available in PMC 2016 August 31.

solely on self-report.


Results of the current study did not support the hypothesis that psychiatric and emergency department employees are more likely to report violence compared with employees on other units.

Comparisons by type of unit (Table 1) revealed that a larger proportion of psychiatric employees were reporters (18.8%) than underreporters (6.2%), but emergency department employees had a larger proportion of underreporters (30.9%) than reporters (28.1%).

However, the researchers did not find a higher likelihood of reporting through the electronic system in any one type of work unit.


Only 32 of the 275 employees (12%) who self-reported violence in the workplace had formally documented the incident via the electronic reporting system.

This finding suggests an overall underreporting rate of 88%.

An earlier study of 47 health care workplaces based entirely on worker self-report found a similar underreporting rate of 79% (Arnetz, 1998).


However, a closer look at the questionnaire data in the current study revealed that more than 45% of the 275 who self-reported violence in the past year (n = 125) used other methods to report the incident.

Those individuals reported the incident verbally to their supervisors (79.2%), via the Compliance Hotline (15.2%), or other means (18.4%).

Although a small number (n = 14, 11.2%) of personnel had also reported through the electronic system, the majority (n = 111) had not.


The only factors significantly associated with informal incident reporting to a supervisor were short job tenure (< 5 years) and working as security staff; both were associated with lower likelihood of reporting.

Sato and colleagues (2013) found that nurses with less work experience were less likely to report aggressive behavior.


Combining all of the alternative methods with electronic documentation, the rate of underreporting dropped to 48%; 143 reporters (32 electronic reporters and 111reporters by other means) leaves 132 of 275 exposed who did not report.

Although this is an improvement over 88%, almost half of the incidents were not reported.


This finding has implications for this and similar health care organizations.

The hospital system in this study has worked for the past decade to establish a centralized electronic system for reporting WPV (Arnetz et al., 2011a, 2011b), and human resource policy mandates reporting of known incidents of violence.

Although employees who verbally report incidents to their supervisors may be fulfilling their responsibility to report, these informal reports may not always be available to upper management for policy decisions.


This is the first study to examine underreporting of WPV among health care workers by linking and comparing individual questionnaire responses (self-report) regarding WPV experiences with actual incident documentation.

However, a number of limitations should be considered.


First, the study was conducted in a single hospital system in one geographic area of the United States, and results may not be generalized to hospitals in other areas.


Second, the response rate on the questionnaire was low (22%).

However, the researchers compared characteristics of respondents with those of non-respondents and found few significant differences.

Moreover, all further analyses were limited to only the cohort of respondents who self-reported WPV (n = 275).


Third, the questionnaire items related to experience of violence were retrospective (past year), and recall bias may have affected the results.

This bias was especially evident in that 63 of the 275 who self-reported violent events (23%) also self-reported having documented at least one incident via the electronic reporting system, when, in fact, only 12 (4%) actually had done so.

It may be that several of these individuals had indeed reported incidents in the electronic system earlier, but not in the past year.

Conversely, of the 212 employees who self-reported that they did not record a violent incident electronically, 20 (9%) actually had.


Finally, the main aim of this study was to better understand the magnitude of underreporting of WPV generally.

Analyses did not examine the type of violence experienced or reported.

As suggested in earlier research (Findorff et al., 2005), it is likely that underreporting of non-physical violence is greater than that of physical violence.


The lack of agreement between employees' survey responses and actual report practices may be due to lack of injury, recall bias, or a lack of motivation to use the central electronic reporting system to report violent events.

Underreporting is a hindrance to determining the actual extent of WPV toward health care workers.

Understanding the magnitude of underreporting and the characteristics of health care workers who are more likely to underreport may provide hospitals with a more accurate estimate of WPV and determine where to focus education, training, and prevention efforts.


The recent study by Arnetz et al. [ 1] concluded that hospital employees underreported incidents of workplace violence mainly because non-victims (witnesses) tended to report incidents far less often than victims.

However, this conclusion is not well supported by the data in their study because of unclear definitions of violence, insufficient separation of the participants into the two groups of victims of violence and witnesses of violence, and plausible alternative explanations for the findings, as explained below.


Of the 446 hospital workers surveyed, a flowchart in Figure 1 in the above study indicates that 62% of them (n = 275) answered “Yes, I was a target of violence within the last year.” Of these 275 respondents, the same flowchart shows that 212 of them answered that “I did not report the violence” in the survey.

However, when asked about reasons for not reporting incidents of violence (Table 3), 30% of the 212 respondents answered that they were neither a target nor witnesses to violence.

The contradicting answers indicate a low reliability for the participants’ determination of whether they were a target of workplace violence.

This problem can be addressed by giving the participants of the survey an option to indicate the nature of the violence experienced and their role (e.g., target, witness).


The low reliability regarding workplace violence may have been due to an unclear definition of violence.

Although the flowchart in Figure 1 documents whether a participant had been a target of violence, the authors state in the “Methods” section that the question was whether the participant had been a target of either violence or aggression with no definition provided for aggression.

The two forms of this question would be expected to produce different responses.

Furthermore, the “Results” section reported that the question was whether the participant had experienced a violent incident, presumably including the witnessing of an incident without being a target.

The confusing definitions of violence could explain the low reliability of the responses, but this is not the only possible explanation for the low reliability of the responses to the same question at two different places in the survey.

Another possible explanation is faulty recall due to the lengthy recall period of 1 year.

Recall periods for injuries or similar events, such as hospital stays, have been shown to be unreliable if more than 3 months prior [2][3].


The study reported that 88% of hospital workers who experienced workplace violence failed to report the incident using the official electronic system, and thus concluded that this finding indicated a need to take action.

The estimate of 88% is likely exaggerated for the following reasons.


First, one incident could have been witnessed by multiple employees but reported only once by one person in the electronic reporting system.

If an employee knows that someone else reported it, they may decide that this is sufficient.

This explanation is supported by the study conclusion that employees tended not to report incidents if they were not the victims of violence themselves.

By the methods used in the study, this explanation would inflate the numbers of employees who failed to report an incident.


Second, the study hospital system only encouraged workers to report via the electronic system, but does not require it. If it is only encouraged, workers cannot be expected to report anything or everything.

It is probably unreasonable to conclude that a failure-to-report problem exists when reporting is not mandated.

This explanation also would be expected to inflate the failure-to-report percentage.


The study by Arnetz et al. has many strengths including an adequate sample size, use of multiple sites within a hospital system, appropriate statistical analysis, clear discussion of workplace violence literature, and a focus on a critical occupational health and safety problem.

Despite these strengths, due to the low response reliability of the key questions and evidence of alternative explanations for the findings, the study conclusion, that hospital employees drastically underreport workplace violence, is not well supported.

Instead, an alternative conclusion, that workplace violence reporting may be acceptable, but the value of the electronic reporting system is low among hospital employees, particularly nurses, is proposed.


We are grateful for the thoughtful comments on our recent article, “Underreporting of Workplace Violence: Comparison of Self-Report and Actual Documentation of Hospital Incidents” [2].

The writer raises issues that are well worthy of discussion.

However, on a number of points, there has been some misunderstanding that we hope this response will clarify.


Conclusion.

The writer states that our article concluded “ . . .

that hospital employees under-reported incidents of workplace violence mainly because non-victims (witnesses) tended to report incidents far less often than victims” [4].

This was not the main conclusion of our article and is factually incorrect.

A closer look at Table 3 (p. 206) reveals no statistically significant difference between reporters and under-reporters with regard to being a target or witness of violence.

Our main conclusions in this article were (a) the majority of hospital workers who experienced violence at work (88%) did not record the incident in the hospital system’s electronic database, (b) nearly half of hospital workers who experienced violence (45%) did report the incident to their supervisors, (c) hospital workers who were injured in a violent incident and/or lost time from work due to a violent incident were significantly more likely to report using the electronic system (Table 4), and (d) workers with less than 5 years job tenure and security staff were significantly less likely to report a violent incident to supervisors (Table 5).


We believed that the juxtaposition of “physical” and “verbal” would suffice as indication to the questionnaire respondents that we were using a broad definition of “violence.” Moreover, in the item asking “What type(s) of violence/aggression did you experience?” the first response alternative was “Verbal aggression (shouting, swearing).

” The writer posits that “the two forms of this question would be expected to produce different responses,” but in truth, we could not be sure that responses would differ.


Workplace violence is subjective; what may be perceived by one worker as “violence” may be perceived by another as lower-level aggression [1].

We recognize that perception may influence reporting, which is why we specifically combined physical violence and aggression in a single question.


Low reliability for the participants’ determination of whether they were a target of workplace violence.

The question on whether the individual had been a target of violence or aggression during the past year was followed by two additional questions: “Who was violent or aggressive towards you?” and “What type(s) of violence/aggression did you experience?” Both items had “I wasn’t a target of violence” as a possible response alternative.

However, questions regarding why employees did not report a violent incident used the response alternative, “I wasn’t a target or witness of violence.

” We believe that this may be the source of the writer’s confusion.


In other words, when asking about experience of violence at work, we asked only whether employees had themselves been the target of violence.

When asking why they had not reported an incident, we included the option that they had not even witnessed such an incident.

The reason the questions were structured this way has to do with hospital system policy, as indicated on page 202 of our article.

Hospital system policy actually mandates that employees report any known incidents of violence, either through the electronic system or to a supervisor.

Policy does not specify that an employee must be a target of violence to report the violent episode; rather, any “known incident of violence” should be reported.


In previous research on this hospital system, we found that incident reporting by third parties not directly involved in the violent incident does occur [3].


Another possible explanation is faulty recall due to the lengthy recall period of 1 year.

We agree with the writer that recall bias is a possible confounder of this study, a point we raised ourselves in the “Strengths and Limitations” section, page 208.

As justification, the survey, part of a large, randomized controlled intervention study, was administered pre-intervention and 1-year post-intervention.

Thus, asking about experience with workplace violence over the course of the previous 12-month period was a deliberate effort on our part to compare self-report with documentation of incidents over the same period of time.


The estimate of 88% is likely exaggerated.

The writer contends that “an incident could have been witnessed by multiple employees but reported only once by one person in the electronic reporting system.

If an employee knows that someone else reported it, they may decide that this is sufficient” [4].

There is nothing in our data to support this statement.

The writer further posits that the statement is supported “by the study conclusion that employees tended not to report incidents if they were not the victims of violence themselves.” As indicated in the point above regarding low reliability, this is a misunderstanding that we hope we have now clarified.


It is probably unreasonable to conclude a failure-to-report problem exists when reporting is not mandated.

This explanation would also be expected to inflate the failure-to-report percentage.

As clarified in the low-reliability point above, reporting acts of known violence is, in fact, mandated by the hospital system.

The writer is correct in that reporting via the electronic system is encouraged but not required.

However, the mandate states that an employee should report either via the system or directly to one’s supervisor.

As we point out on page 202, unit supervisors also have a mandate to report all violent events of which they are notified into the electronic system within 24 hours from the end of the shift during which they received the notification.

Thus, theoretically, incidents should ultimately be entered into the electronic database.

Our study indicates that electronic database reporting occurs in only 12% of cases.


. . . the value of the electronic reporting system is low among hospital employees, particularly nurses.

We can agree that the electronic reporting system is underused for reporting incidents of workplace violence, and among nurses, there were more under-reporters (62.6%) than reporters (40.6%, Table 1).

However, other, more far-reaching implications of this study’s findings exist.

As we stated on page 208, although employees who report verbally to their supervisors may be fulfilling their responsibility for reporting, the informal reports may not be reaching upper management for policy decisions.

A hospital system or other health care organization can only develop prevention strategies based on available data.

By implementing the electronic system, this particular hospital system has made a concerted effort to facilitate the employee reporting process.

Our data indicate that the system is underused.


While PBHs can be generated at the QCD epoch [14] (for a recent analysis see Ref.

[ 15]) or during phase transitions [16], a popular mechanism for the formation of PBHs is within the so-called "spiky" scenario in which PBHs are originated from the enhancement of the curvature power spectrum arXiv:1801.09415v2 [astro-ph.

CO] 12 Mar 2018 below a certain length scale due to spiky features [17-19].

Such an enhancement can be produced either within single-field models of inflation, see for instance Refs.

[ 20-22] for some recent work, or through a spectator field which does not contribute significantly to the total curvature perturbation on large observable scales (which is ultimately responsible for the anisotropies in the cosmic microwave background anisotropies), but whose unique role is precisely to enhance the power spectrum at small scales, see for example Refs. [ 23-26].

This spectator field can be even the Higgs of the Standard Model [27, 28].


Once the power spectrum of the curvature perturbation has been enhanced during inflation from its ∼ 10 −9 value at large scales to ∼ 10 −2 on small scales, and after the fluctuations have been transferred to radiation during the reheating process after inflation, PBHs may form from sizeable fluctuations in the radiation density field if they are able to overcome the resistance of the radiation pressure.


A perturbation of fixed comoving size may not start to collapse till it re-enters the cosmologi- cal horizon.

Therefore, the size of a PBH at formation is related to the horizon length when the corresponding perturbation enters the horizon and collapses.

Fluctuations collapse immediately af- ter horizon re-entry to form PBHs if they are sizeable enough.

Indicating by ζ the gauge-invariant comoving curvature perturbation with power spectrum P ζ , one can define the smoothed version of it ζ R ( x) = d 3 y W (| x − y|, R) ζ( y) (1.1) and its variance σ 2 (R) = d 3 k (2π) 3 W 2 (k, R)P ζ (k), (1.2) where the smoothing out is operated through a window function W (k, R) where one has to select the radius as the comoving horizon length R H = 1/aH. A given region collapses to form a PBH if this variance is larger than a critical value ζ c (0.05 − 1).

The actual value depends on the equation of state w of the fluid when the perturbation re-enters the horizon [29]


For radiation w = 1/3 and one obtains ζ c 0.086.

Larger values are also used in the literature [30-32].

A common value is ζ c 1.3 and we will adopt it as representative from now on.

Notice that one could use also a better criterion [30] based on the density contrast (during the radiation era and on comoving slices) ∆( x) = (4/9a 2 H 2 )∇ 2 ζ( x) for which the critical value is ∆ c 0.45 [33] 1 .

At any rate, our formulae will be valid for both ζ( x) and ∆( x) and their corresponding power spectra on small scales must be rather large if a significant number of PBHs have to form.

The mass of a PBH at formation may be directly related to the number N of e-folds before the end of inflation when the corresponding density fluctuation leaves the Hubble radius M (m 2 P /H)exp(2N ) [35], where m P is the Planck mass.


Now, defining ν(M ) = ζ c σ(M ) , (1.6) the Gaussian mass fraction β prim (M ) for ν ∼ > 5 can be well approximated by


Since PBHs are generated through very large, but rare fluctuations, their mass fraction at formation is extremely sensitive to changes in the tail of the fluctuation distribution and therefore to possible non-Gaussianities in the density contrast fluctuations [19, 40-48].

This implies that non-Gaussianities need to be accounted for within the spiky scenario as they can alter the initial mass fraction of PBHs in a dramatic way.


We are certainly not the first ones to make this point.

For instance, the presence of a scale-invariant local non-Gaussianity, which is nowadays severely constrained by CMB anisotropy measurements [49], can significantly alter the number density of PBHs through mode coupling [50, 51].

In fact here we are rather referring to that non-Gaussianity which is almost inevitably generated at the same small wavelengths where the density perturbations are sizeable and that in the literature is often disregarded, most probably given the associated technical difficulties.


To have the feeling of the level of non-Gaussianity associated to the large perturbations giving rise to the PBHs, let us consider the second-order gauge-invariant comoving curvature perturbation ζ 2 .

During inflation, when the perturbations have already left the Hubble radius, it satisfies the equation [52]ζ 2 = − H (ρ 0 + P 0 ) δP 2 | ρ − 2 (ρ 0 + P 0 ) δP 1 | ρ − 2(ρ 0 + P 0 )ζ 1 ζ 1 , (1.8) where ρ 0 and P 0 are the background energy and pressure densities, δP | ρ is the gauge-invariant non- adiabatic pressure perturbation (on uniform density hypersurfaces) and ζ 1 is the linear gauge-invariant comoving curvature perturbation.


In single-field models the non-adiabatic pressure perturbation van- ishes and the curvature perturbation ζ is conserved at any order in perturbation theory on super-Hubble scales, consistently with Eq. ( 1.8).

Indeed, in single-field models the perturbations are generated at Hubble crossing and Eq. ( 1.8) is not useful to estimate the level of non-Gaussianity associated to the fluctuations responsible for the PBHs and one has to resort to a different way of estimating the level of non-Gaussianity.


However, in the case of spectator fields, the curvature perturbation is not conserved on super-Hubble scales since the non-adiabatic pressure perturbation is not zero.

In such a case, Eq. ( 1.8) signals that the same sizeable linear perturbations which will be responsible for the generation of the PBHs upon horizon re-entry inevitably give rise to sizeable second-order perturbations and therefore to non-Gaussianity, that is ζ 2 = O(1)ζ 2 1 barring cancellations.


This simple argument shows that the level of non-Gaussianity in the large, but rare fluctuations associated to the PBHs nay be not insignificant.


The goal of this paper is to discuss the impact of non-Gaussianity on the initial PBH mass fraction.

In particular,


1. we follow closely Ref.

[ 53] and use a path-integral formulation to provide the probability that the density contrast is larger than a critical value within the horizon at formation time.

The expressions are not original, but maybe known more in the community dealing with halo bias.

They are exact and no approximations are taken;


2. discuss the fine-tuning needed to achieve for the non-Gaussianity to play no role in the deter- mination of the PBH mass function;


3. compute in two popular classes of spiky models the impact of non-Gaussianity, showing that neglecting it leads to blunders in the estimation of the PBH mass fraction;


4. discuss the impact of non-Gaussianity on the clustering of PBHs.


The paper is organised as follows.

In section II we describe the technicalities based on the path- integral approach which allow to write exact expressions for the various quantities of interest in the presence on non-Gaussianity.

In section III we introduce the notion of fine-tuning associated to the non-Gaussianity and in relation to the primordial PBH mass fraction.

Sections IV and V contain two illustrative examples and section VI touches the notion of clustering and non-Gaussianity.

Finally, conclusions are contained in section VII.


This section addresses the calculation of the exact non-Gaussian PBH abundance through a path- integral approach.

It has the virtue of not making any simplifying assumption or approximation, but at a price of being rather technical.

The uninterested reader can jump directly to the final result in Eq. ( 2.28) or its simplified version (2.32).

We follow closely Ref.

[ 53] which we reproduce the results of.

In this sense this section does not contain any new finding, but maybe results are derived in a more transparent way.


Consider the overdensity field ζ( x) with a generic probability distribution P [ζ( x)].

The partition function Z[J] in the presence of an external source J( x) is given by Z[J] = [Dζ( x)]P [ζ( x)]e i d 3 xJ( x)ζ( x) , (2.1) where the measure [Dζ( x)] is such that


The functional Taylor expansion of the partition function Z[J] in powers of the source J( x) specifies the correlators, while the corresponding expansion of W [J] = ln Z[J] generates the connected correlation functions, which we will denote by We have already introduced the smoothed curvature perturbation over a scale R in Eq. ( 1.1).

In a similar manner, we define smoothed connected correlation function ξ (N ) R ( x 1 , · · · , x N ) as ξ (N ) R ( x 1 , · · · , x N ) = ξ (N ) R ( y 1 , · · · , y n ) N i=1 d 3 y i W (| x i − y i |, R), (2.4) so that ξ (2) R ( x, x) = ξ (2) R (0) = σ 2 R , (2.5) is the variance over the scale R. For later use, we also define the normalised correlators


Let us also recall that the functional W [J] is written in terms of the connected correlation functions ξ (N ) as


We now wish to characterise the threshold statistics by requiring that ζ R exceeds a certain threshold ζ c , which we can write as ζ R > νσ R .

Then the peak overdensity ρ ν,R ( x) is defined as ρ ν,R ( x) = Θ(ζ R ( x) − νσ R ), (2.9) where Θ(x) is the Heaviside step-function.

The joint probability that in all points, ( x 1 , · · · , x N ), the overdensity is above the threshold is given by


Since dΘ( x)/dx = δ( x) and 2πδ( x) = ∞ −∞ dφ e iφx , the Θ-function may be expressed as Θ( x) = ∞ −x da ∞ −∞ dφ 2π e iφa , (2.11) and therefore,


where J is defined as J( x) = N i=1 J i with J i ( x i , x) = φ i W (| x − x i |, R), and in the third line in Eq. ( 2.12) we have redefined a i → a i /σ R .

Using Eq. ( 2.8) we may express ln Z[J] as


We may put the last expression in a more explicit form using Eq. ( 2.5)


To calculate the expression (2.18), we have used the formula


where H m−1 are Hermite polynomials and Erfc(x) = √ 2/π ∞ x exp(−t 2 )dt is the complementary error function, we find that Eq. ( 2.21) can be re-expressed as Π (N ) ν,R ( x 1 , · · · , x N ) = ∞ =0 1 !

   { 1 ,··· , N }∈P ,N ! n 1 ! · · · n N !

Ξ { 1 ,··· , N } N r h r (ν)    , (2.23) where Ξ { 1 ,··· , N } denotes the sum of Ξ( x i 1 , · · · , x i )'s at 1 , · · · N points.

For N = 3 and = 4, the possible partitions are (4, 0, 0), (3, 1, 0), (2, 2, 0), (2, 1, 1) and then we have for example,


We are finally ready to collect the results to find the probability for a single density contrast to be above the threshold.

It can be directly calculated by evaluating the one-point statistics


where the argument (0) means that all the correlation functions are computed at the same point, Ξ n (0) = Ξ( x, x, · · · , x), (2.29) and Ξ( x 1 , x 2 , · · · , x n ) is given by Eq. ( 2.20) with ξ (2) R omitted.

For instance, we have


We draw the attention of the reader to the fact that the expression (2.28) is exact and no approxima- tions have been taken along the way.

Nevertheless, it simplifies in the case of large threshold ν 1.

In this limit, using the asymptotic behaviour (ν 1) of the Hermite polynomials and the complementary error function


we find that Eq. ( 2.28) reduces to


where the argument (0) means that all the correlation functions are computed at the same point.

If the smoothing radius is taken to be R H , this expression becomes the mass fraction β prim (M ).

Apart from the limit ν 1, it is exact as it contains all the non-Gaussian correlators for which no assumption needs to be taken.

If one wishes to relax even the assumption of large ν, one can resort to Eq. ( 2.28).


After this technical detour, we are ready to ask under which circumstances non-Gaussianity alters the predictions of the primordial abundance of black holes in a significant way.

To answer this question one of course should look at her/his own preferred model and estimate the amount of non-Gaussianity through the non-vanishing correlators with order larger than two.

However, one can take inspiration by what is routinely done in high energy physics and introduce the notion of fine-tuning, that is how much sensitive is a given observable on the parameters it depends upon.

For instance, in supersymmetric models, where the phenomenon of electroweak symmetry breaking is induced radiatively by the running with energy, one may compute the response of an observable such as the mass squared of the Z-boson m 2 Z to a change of a parameter of the model defined at some high energy [55].


In full similarity, we can ask ourselves how sensitive is β prim (M ) to the various cumulants, defined by the relations


We may therefore define the fine-tuning ∆ n to be the response of the PBH abundance to the intro- duction of the n-th cumulant


Given the particular structure of the non-Gaussian correction to the PBH abundance, one can im- mediately understand the physical significance of this fine-tuning parameter.

Indeed, in the presence of the n-th cumulant, one can express the new non-Gaussian PBH abundance (2.32) in terms of the Gaussian abundance


This implies that the PBH abundance is exponentially sensitive to the non-Gaussianity unless ∆ n is in absolute value smaller than unity


Inspecting Eqs. ( 1.7) and (2.32), we see that


This tells us that non-Gaussianity alters exponentially the Gaussian prediction for the PBH abundance unless


To investigate how restrictive this condition is, we will analyse two representative classes of spiky models, one based on a single-field and the other based on the presence of a spectator field.


In single-field models of inflation the power spectrum of the comoving curvature perturbation is given by [56] P ζ (k) = H 2 8π 2 m 2 P 1 k 3 , (4.1) where = −d ln H/dN is one of the slow-roll parameters and N is the number of e-folds till the end of inflation.

Since the generation of PBHs requires the jumping of the value of the power spectrum of about seven orders of magnitude from its value on CMB scales, without even specifying the single- field model of inflation, one may conclude that there must be a violation of the slow-roll condition |∆ ln /∆N | ∼ > O(1) [32].

In particular, this happens when the inflaton field goes through an inflection point in the scalar potential, see Fig. 1, thus producing a sizeable resonance in the power spectrum of the curvature perturbation.

Correspondingly, slow-roll conditions are broken and the spike in the curvature perturbation may lead to a copious production of PBHs, see for instance Refs. [ 20, 22] for some recent proposals.


When the inflaton experiences a plateau in its potential, since must be extremely small, a short period of Ultra Slow-Roll (USR) is achieved during which the equation of motion of the inflaton background φ 0 reduces to [57-62]φ 0 + 3Hφ 0 0, (4.2) and the piece dV (φ 0 )/dφ 0 is negligible so that there is an approximate shift symmetry in the inflaton field.

The slow-roll parameter η =˙ H = 2φ 0 φ 0 H + 2 −6 (4.3) is therefore sizeable, signalling the breaking of the slow-roll conditions.

During such a phase of USR inflation, = 3 2φ 2 0 V (φ 0 ) ∼ a −6 , (4.4) and the comoving curvature perturbation increases as P ζ ∼ e 6∆N , (4.5) where ∆N parametrises the duration of the USR phase.

Thanks to the approximate shift symmetry as well as the dilation symmetry, one can characterise the bispectrum of the perturbations generated during the USR phase.

The non-Gaussianity turns out to be completely local (δφ is basically a massless scalar field on the de Sitter background in the limit ≪ 1 and non-Gaussianity is generated after Hubble radius crossing) [59, 60, 63]


The corresponding skewness reads


where we have neglected the effect of the window functions in the momentum integrals 2 .

Since the condition (3.6) for non-Gaussianity to play no significant role is S 3 ∼ < 0.06 σ 2 R H 0.02 1.3 ζ c 3 , (4.8) where we have normalised σ 2 R H to the value necessary to get the right abundance of dark matter from PBHs [32] (neglecting accretion), we see that the Gaussian estimate on the PBH mass fraction is hardly trustable.

Of course this result strongly depends on the values of σ 2 R H and ζ c , but it signals a potential danger.


In fact, one can do better at this stage.

We can use the δN formalism [66-68] to evaluate the higher-order cumulants.

In the δN formalism the scalar field fluctuations are quantized on the flat slices and related to the comoving curvature perturbation ζ through the relation ζ = δN .

During the USR phase and in the case of a sharp transition back to the slow-roll phase, one obtains [59] N (φ 0 ,φ 0 ) = 1 3 ln φ 0 φ 0 + 3H(φ 0 − φ e ) , (4.9) where φ e is the value of the field at the end of the USR phase.

Notice in particular that one has to retain the dependence onφ 0 since slow-roll is badly violated.

Since δφ is constant to good accuracy, one can neglect δφ and obtain


where ζ 1 ( x) is the linear curvature perturbation.

If one considers the contribution to the cumulant S n coming from the connected piece of the form


This contribution shifts the Gaussian exponential in the mass fraction in the expression (2.32) from (in this expression one should intend the variance σ 2 R H as the renormalized one once the non-Gaussian contributions are as well summed up [40, 65])


This is not at all an insignificant effect for ν 1.

For instance one can ask what is the impact on the PBH abundance at formation.

We follow Ref.

[ 32] and we set β prim (M ) such that it corresponds today to a totality of the dark matter in the form of PBHs.


Choosing ζ c 1/3, a Gaussian probability would give σ R H ∼ > 0.036, while the non-Gaussian result (4.14) would deliver a result smaller than about ten orders of magnitude.

This shows the large sensitivity of the non-Gaussian result to the cosmological observables.

Of course, this is not the final result as there are other contributions to the cumulants we have not accounted for.


For instance, S 4 receives a connected contribution from the correlator ζ 1 ( x)ζ 1 ( x)ζ 2 1 ( x)ζ 2 1 ( x) .

We have not been able to resum all the pieces.


Despite their incompleteness and the fact that the details have to be worked out model by model, our findings indicate that non-Gaussianity plays a fundamental role in determining the final PBH abundance.


Notice that we have been working under the assumption of a sudden transition between the USR phase and the slow-roll subsequent stage.

It has been recently noticed that the non-Gaussianity is sensitive to the details of the transition from USR phase to the standard slow-roll phase [64].

The skewness takes the generic value S 3 9(4(η V − 3)η V + h 2 + 4η V h)/(2η V + h − 6) 2 , where h = 6 √ V /π e , π = dφ 0 /dN and V and η V = V /3H 2 are evaluated at the slow-roll phase.

For |h| 1 one recovers the sharp transition case, while for a smooth transition or a sharp transition, but small h, one finds the value S 3 η V .

Said in other words, our results show that a smooth transition is preferable in order to have a trustable Gaussian estimate.


The second example we wish to illustrate falls in the category of spiky models where the comoving curvature power spectrum is enhanced at small scales through a spectator field [23].

The idea is that there is a curvaton-like field σ [69] with quadratic potential V (σ) = 1 2 m 2 σ 2 , (5.1) such that the total curvature perturbation during inflation reads (in the flat gauge) ζ = H δρ ρ =ρ φ ρ ζ φ +ρ σ ρ ζ σ , ζ φ,σ = H δρ φ,σ ρ φ,σ , (5.2) where ζ σ is the curvaton perturbation and ζ φ is responsible for the anisotropies in the CMB anisotropies on large scales.

The two contributions are assumed to be comparable at a scale k c Mpc −1 .

The curvaton is in fact an axion-like field.

It is related to the phase θ of a complex field whose modulus is the inflaton field φ 0 with potential V (φ 0 ) = 1 2 λH 2 φ 2 0 , (5.3) with 1 < λ ≤ 9/4.

The inflaton rolls relatively fast towards its minimum φ * which is reached when the wavenumber k * is exiting the Hubble radius and the curvaton field becomes well-defined, σ( x) = φ * θ( x).

For scales k < k * , when the inflaton is still rolling down its potential, the fluctuations of the curvaton inherit a strong scale-dependence, δθ k (H/2πφ 0 (t k )), where t k is the time at which the wavelength 1/k exits the Hubble radius.

The corresponding spectral index is given by n σ − 1 3 − 3 1 − 4 9 λ, (5.4) and one can obtain a blue curvaton power spectrum, n σ O(2 − 4) [23].


After inflation, the curvaton oscillates around the minimum of its potential and it decays into radiation.

If we denote by r the ratio of the curvaton energy density ρ σ over the radiation energy density ρ γ , at small scales the linear curvature perturbation is dominated by the curvaton and becomes


The perturbation is nevertheless getting a non-Gaussian contribution.

Restricting ourselves to the quadratic potential (5.1), the curvature perturbation receives second-order contributions from the (δσ( x)) 2 piece in δρ σ .

A standard computation leads to [70-72]


Neglecting for the moment the window functions, the corresponding skewness is S 3 6 r 3 2 − r 2 , (5.7) which is a decreasing function of r with S 3 = 3 at r = 1.

Imposing again S 3 ∼ < 0.06 σ 2 R H /0.02 (1.3/ζ c ) 3 , one finds solutions only for sufficiently small thresholds or large variances.

If we consider higher-order terms in the curvaton axion-like potential, higher-order cumulants are generated, but they will depend on other parameters, such as the quartic coupling ∼ m 2 /φ 2 * and the background value σ 0 .

We do not expect cancellations to operate and make the Gaussian mass fraction of PBHs trustable.


To estimate the impact of non-Gaussianity, we can use alternatively the density contrast (during the radiation phase) ∆( x) = (4/9a 2 H 2 )∇ 2 ζ( x) for which ∆ c 2 √ 2ζ c /9 0.4(ζ c /1.3).

The corresponding skewness reads


The numerical results are shown in Fig. 2 which makes explicit that the Gaussian evaluation of the PBH abundance does not suffice.

One obvious improvement will be to evaluate the one-loop variance induced by the non-Gaussianity δσ 2 ∆ (R H ) = 2 4 9 2 1 r 2 3 2 − r 2 2 d 3 k (2π) 3 (kR H ) 4 W 2 (k, R H ) d 3 p (2π) 3 P ζ (p)P ζ (| k − p|), (5.9) where the power spectra are of course the tree-level ones.

This improvement will however require an improved computation of the skewness and so on.

We again follow Ref. [ 32] and ask what is the impact of non-Gaussianity if we demand that the primordial PBH abundance corresponds to the totality of the dark matter abundance today.


Let us minimise the influence of non-Gaussianity by setting r = 1 and S 3 3.

Under these circumstances, δσ 2 ∆ (R H ) is still much smaller than the tree-level σ 2 ∆ (R H ), and the Gaussian approach would give σ 2 ∆ (R H ) ∼ > 0.02 [32].

This is in sharp contrast with our condition (3.6) on S 3 which would deliver S 3 ∼ < 0.06.

A value of S 3 would deliver a dark matter abundance smaller by almost twenty orders of magnitude.

This again indicates the sensitivity of the cosmological predictions to the non-Gaussianity.


The PBHs created in the radiation era and originated from spikes in the comoving curvature perturba- tion are highly clustered [34].

Using the peak theory model of bias, one can compute the PBH n-point correlation functions.

For Gaussian fluctuations one can follow Ref. [ 73] and show that the connected two-point correlator of PBHs is given by


One can easily see that the correlator ξ (2) ν,R H ( x 1 , x 2 ) increments for higher values of ν.

In other words, since PBHs are generated by large fluctuations and their mass fraction is exponentially suppressed for ν 1, the fewer there are of them, the more clustered they are.

The technique explained in Ref.

[ 53] allows also to compute the correlators of the density contrasts above a given threshold for a non-Gaussian statistics.

Let us for instance consider the case of W ( x, R) = δ (3) ( x).

It follows immediately that in the large threshold limit the joint probability is Π (N ) ν,R ( x 1 , · · · , x N ) = e −N ν 2 (2πν 2 ) N/2 N i=1 Z[J i ], (6.2) which, by using Eq. ( 2.32) can be written as


Therefore, the N -point peak disconnected correlation functions ξ (N ) disc,ν,R ( x 1 , · · · x N ) defined as ξ (N ) disc,ν,R ( x 1 , · · · , x N ) = N i=1 ρ ν,R ( x i ) ρ ν,R − 1, (6.4) turn out to be


For instance, the non-Gaussian two and three-point peak connected correlation functions read respec- tively


As already stressed in Ref. [ 53] (see also Ref. [ 54]) for the validity of these equations, correlations do not need to be small either with respect to unity or with respect to the variance on the smoothing scale.

The two-point correlator can be rewritten as


The power spectrum in momentum space and over the normalization volume V will read


where the lower cut-off of the integral is due to the finite size of the PBHs.

One can expand in powers the exponentials and obtain


For instance, for the class of single-field model generating PBHs at inflection points described in section IV one finds that the Fourier transform of ν 3 /σ 3 R H ξ (3) R H ( x 1 , x 2 , x 2 ) is given by


that is a factor ∼ 6ζ c times the Gaussian contribution.

One sees again that the non-Gaussian correc- tions can alter the clustering properties significantly.

Since clustering influences the merging of PBHs, this could have a severe impact on the subsequent cosmological evolution of PBHs.

For instance by changing the limits on initial PBH abundance or, as merging enhances the abundance of more mas- sive and long-lived PBHs, by making PBHs the seeds of supermassive BHs [74, 75].

Our preliminary analysis above suggests that non-Gaussianity may not be disregarded in these considerations.

We will defer the study of this subject in a forthcoming publication.


The idea that PBHs may be the dark matter in our universe is a fascinating one.

Equally exciting is the fact that PBHs may owe their origin to large and rare fluctuations generated during inflation, i.e. during the phase of accelerated expansion which might be ultimately responsible also for the large-scale structures we see around us.

If this is the mechanism nature has chosen to permeate the universe with dark matter, it is vital to understand in details the quantitative predictions within a given set-up.

In this paper we have discussed the role of non-Gaussianity in determining the PBH mass fraction at formation time and their clustering properties.

We have done so by discussing a path-integral procedure which leads to exact expressions for the quantities of interest and by estimating the impact of non-Gaussianity in a couple of models.

Our findings show that non-Gaussianity should be taken into account when considering a model for PBH production and the corresponding non-Gaussianity parameters should be evaluated in order to gauge if PBHs are interesting candidates for dark matter and when analysing their clustering properties and subsequent merging and accretion.

Also, it will be interesting to investigate the impact of the non-Gaussianity on the production of the gravity waves from inflation generated by the same large perturbations originating the PBHs [8] and on the limits coming from CMB µ-distortions [76].


We consider mechanisms for producing a significant population of primordial black holes (PBHs) within string inspired single field models of inflation.

The production of PBHs requires a large amplification in the power spectrum of curvature perturbations between scales associated with CMB and PBH formation.

In principle, this can be achieved by temporarily breaking the slow-roll conditions during inflation.

In this work, we identify two string setups that can realize this process.

In string axion models of inflation, sub-leading non-perturbative effects can superimpose steep cliffs and gentle plateaus onto the leading axion potential.

The cliffs can momentarily violate the slow-roll conditions, and the plateaus can lead to phases of ultra slow-roll inflation.

We thus achieve a string motivated model which both matches the Planck observations at CMB scales and produces a population of light PBHs, which can account for an order one fraction of dark matter.

In DBI models of inflation, a sharp increase in the speed of sound sourced by a steep downward step in the warp factor can drive the amplification.

In this scenario, discovery of PBHs could indicate non-trivial dynamics in the bulk, such as flux-antibrane annihilation at the tip of a warped throat.


In the early 1970s, Stephen Hawking proposed that overdense inhomogeneities in the very early Universe could gravitationally collapse to form Primordial Black Holes (PBHs) [1, 2].

Assuming the Planck length as the minimal possible Schwarzchild radius, gives a lower limit for black holes masses of about 10−5g. Moreover, this mechanism – in contrast to the astrophysical process from dying stars – allows for populations of black holes spanning a vast range of masses, from the Planck scale to supermassive and beyond.

Constraints on PBHs abundances continue to improve, but leave viable windows, especially when astrophysical uncertainties are taken into account (see [3, 4] for recent reviews).

PBHs could provide a significant fraction – or indeed all – of the mysterious Dark Matter that dominates cosmic structures in the present day Universe.

Stimulated both by the absence of signatures for well-motivated particle candidates for Dark Matter and by the first detection of gravity waves from merging black holes, this idea has recently taken new flight.

The principal modern process for PBH production is the collapse of curvature perturbations generated during early Universe cosmic inflation, when the relevant scales re-enter the horizon during the radiation era [5].

However, the big challenge here is that the amplitude of these inflationary curvature perturbations at the large scales probed by the CMB has been constrained by observations to be tiny, As ∼ 10−9 at the pivot scale k = 0.05 Mpc−1.

In order for inflationary perturbations to lead to a significant PBH population there must be some mechanism at work that enhances the power spectrum to around As ∼ 10−2 at some smaller scale.

For example, a large enhancement at scales around k ∼ 1012 − 1014 Mpc−1 would lead to PBHs masses in the window 10−17M(cid:12) (cid:46) MP BH (cid:46) 10−13M(cid:12), which could be a dominant component of Dark Matter consistently with current observational constraints1.


Many recent works have explored how such an enhancement could be achieved.

As current CMB observations are consistent with the simplest, single field slow-roll models, much of the focus has been to explore scenarios producing PBHs in single field inflation [7–11].

A possibility is that the inflaton rolls down a slow-roll plateau, probed by the CMB observations, followed by a near-inflection point which leads to a phase of ultra slow-roll.

Roughly, the extreme flattening of the inflationary potential around the inflection point leads to an enhanced amplitude of scalar perturbations.

Whilst such a potential can certainly be manufactured within effective field theory, ultimately its shape – and robustness against quantum corrections – would have to be explained within a fundamental theory, like string theory.

For example, an attempt towards embedding this model within string theory via fine-tuned string-loop corrections in fibre inflation has been made very recently in [12], although it has not yet proved possible to fine-tune a setup such that cosmological perturbations both match the CMB and produce significant PBH Dark Matter.


The purpose of this paper is to identify well-motivated string mechanisms that can lead to an amplification of curvature perturbations and thus significant PBH production in the early Universe. To do so, it is helpful to recall the physical conditions underlying this amplification, first discussed by Leach et al. [ 13, 14]2.

In slow-roll inflation, the equation that governs the curvature perturbation takes the form of a damped harmonic oscillator.

Solutions for super-horizon modes correspond to a linear combination of a constant and a decaying mode.

However, if the slow-roll conditions are broken, then the friction term in the damped harmonic oscillator equation can become a driving term, so that the decaying mode becomes a growing mode, and enhances the curvature perturbation on scales beyond the horizon.

This enhancement continues until the slow-roll conditions are restored.

( A similar mechanism has been used to build single field models with large squeezed non-Gaussianity [16].)