Hawking radiation and analogue experiments: A Bayesian analysis

Highlights

• A Bayesian analysis of analogue experiments on Hawking radiation is provided.

• A Theorem proves confirmation via an appeal to `universality arguments'.

• Two further theorems study further quantitive implications.

Abstract

We present a Bayesian analysis of the epistemology of analogue experiments with particular reference to Hawking radiation. First, we prove that such experiments can be confirmatory in Bayesian terms based upon appeal to ‘universality arguments’. Second, we provide a formal model for the scaling behaviour of the confirmation measure for multiple distinct realisations of the analogue system and isolate a generic saturation feature. Finally, we demonstrate that different potential analogue realisations could provide different levels of confirmation. Our results provide a basis both to formalise the epistemic value of analogue experiments that have been conducted and to advise scientists as to the respective epistemic value of future analogue experiments.

Introduction

Recent years have seen a proliferation of experiments designed to probe the phenomenon of Hawking radiation via analogue black hole systems. Reports on these experiments include claims of observation of classical, thermal aspects of Hawking radiation in an analogue white hole created using surface water waves (Weinfurtner et al., 2011, Weinfurtner et al., 2013) and observation of quantum entanglement across an analogue horizon, as is expected from Hawking radiation, using a Bose-Einstein Condensate (BEC) (Steinhauer, 2016a). Despite these successes, an uncontested and repeated experimental demonstration of the full analogue Hawking phenomenon is still to be provided.¹ The not fully settled experimental status of analogue Hawking radiation not withstanding, there is a clear need for methodological appraisal of what, in principle, can be established about black hole Hawking radiation based upon analogue experiments. The most pressing questions is whether or not there are circumstances in which analogue experiments can be taken to provide inductive support for conclusions about astrophysical black holes. Can analogue experiments provide ‘experimental confirmation of Hawking's prediction’ (Jeff Steinhauer quoted in Haaretz (Efrati 2016)), or are they simply ‘amusing feat[s] of engineering’ that ‘won't teach us anything about black holes' (Daniel Harlow quoted in Quanta (Shmahalo 2016)).

In this paper we will substantially extend previous philosophical work characterising analogue black hole experiments as a form of ‘analogue simulation’ (Dardashti, Thébault, & Winsberg, 2017; Thébault, 2019) via application of Bayesian confirmation theory. In that previous analysis, emphasis was placed upon the qualitative claim that certain ‘universality arguments’ can be used to link evidence about the ‘source’ dumb hole system to the ‘target’ black hole system. The results of this paper are quantitive in nature and licence normatively valuable conclusions regarding the structure of such inferences.

We first present a Theorem that demonstrates how the confirmation claim can be qualitatively characterised in Bayesian terms. The role of the universality arguments is understood in terms of support for a background assumption that is common between the source and target models. This means that there is a binary variable that can be assumed to be positively correlated with the empirical adequacy of both the source and target models. Evidence in favour of the model of the source system can thus be used to make inferences about the target system. Although not in-and-of-itself a validation of the argument for confirmation via analogue simulation, the Bayesian analysis demonstrates the internal consistency of the informal arguments given in (Dardashti et al., 2017). Furthermore, the formal model provides a powerful evaluative and heuristic tool for the further analysis of the structure of the inferences involved in cases of analogue simulation. In particular, in the remains of the paper, we present two further results that we expect will be of interest to contemporary analogue black hole experimentalists.

Our second principal result relates to the behaviour of the confirmation measure in the context of experimental realisations of the analogue system in different types of media. This is of particular relevance to contemporary analogue black hole research where a diverse range of analogue realisations of Hawking radiation are being pursued: e.g., surface waterwaves, BECs, superfluid helium-3, moving optical media. The immediate question in this context is how the number of distinct types of analogue system one constructs relates to the confidence one should have in the astrophysical effect. The second key result of this paper is a formal model for ‘multiple source’ analogue simulation displaying the generic feature of ‘saturation’ in confirmatory power with an increase in the number of sources. Significantly, the saturation in confirmation indicates that, under plausible assignments of priors, even an extraordinarily large range of diverse analogue experiments will not lead to conclusive confirmation of astrophysical Hawking radiation. This is in tune with the scientific intuition that there is a limit to what can be learned about astrophysical Hawking radiation via analogue experiments.

Finally, and perhaps most intriguingly, there is the question of whether different potential analogue realisations could provide different levels of confirmation. Would we learn more about astrophysical black holes from an analogue experiment based upon liquid helium or BECs? The third key result of the paper is a Theorem proving that analogue experiments in which we are more confident about the fundamental physics (e.g. BECs) teach us less about the target system than those about which we are less confident (e.g. superfluid helium-3). Our results thus provide a basis to both formalise the epistemic value of analogue experiments that have been conducted and to advise scientists as to the respective epistemic value of future analogue experiments. As such, our work demonstrates the enduring value of the Bayesian framework as a tool for analysing the protean forms of scientific inference.