Forschung
Die folgenden Studien sind ausgewählte Beispiele von Forschungsprojekten, die ich als Erstautor veröffentlicht habe.
Local Geometric Bounds on Generalized Entropy Evolution along Null Horizons
We develop a local and covariant framework for constraining the evolution of generalized entropy along null horizons, combining classical geometric methods with constraints from quantum field theory. Building on the quantum focusing conjecture (QFC), which restricts entropy evolution along null directions, we derive a Raychaudhuri-type differential inequality for the generalized expansion that makes its local dependence on expansion and shear explicit. The resulting relation, $\frac{d\Theta}{d\lambda} \le -\theta \Theta + \tfrac{1}{2}\theta^2 - \sigma^2$, reveals a direct interplay between expansion and shear in controlling entropy flow: shear contributes negatively and yields a monotonic suppression of the generalized expansion in shear-dominated regimes, while expansion provides a competing geometric source term. This structure provides a local geometric formulation consistent with the QFC and clarifies its interpretation as a constraint on entropy evolution. We further show that quantum extremal surfaces correspond to configurations characterized by $\Theta=0$, whose local stability properties are governed by the same geometric data. In addition, we derive a bound on the exponential separation of nearby null generators, indicating that the same combination of expansion and shear also controls geometric instability. Our results provide a local geometric refinement of entropy bounds in semiclassical gravity and establish a direct connection between null geometry, quantum energy conditions, and entropy flow, offering a local geometric perspective on horizon thermodynamics beyond global formulations.
Table: Analogy between classical geometric focusing in general relativity and entropy dynamics along black hole horizons. The expansion $\theta$ of a geodesic congruence corresponds to the generalized expansion $\Theta$, which governs the local rate of change of the generalized entropy; the Raychaudhuri equation for $\theta$ therefore has a direct counterpart in the entropy Raychaudhuri inequality for $\Theta$. In this way, classical focusing of geodesics maps to a dissipation-like behavior of entropy flow, while standard energy conditions in the classical theory are replaced by the Quantum Null Energy Condition and the Quantum Focussing Conjecture in the quantum regime. Finally, trapped surfaces in classical geometry encode the analogue of quantum extremal surfaces, which describe stationarity conditions for the generalized entropy and provide a quantum‑corrected notion of horizon structure.
More information:
Bertram, E., Physical Review D, 2026
Local Geometric Bounds on Generalized Entropy Evolution along Null Horizons
From Triadic Interactions to Kolmogorov Scaling: A Deterministic, Scale-Resolved Formulation of Energy Flux
We develop a deterministic, scale-resolved formulation of energy transfer in the three-dimensional incompressible Navier–Stokes equations based on an explicit triadic decomposition of the nonlinear term in Fourier space. Using a systematic dyadic localization of the velocity field, we derive an exact representation of the nonlinear energy flux across scales and organize it in terms of interactions between well-defined scale components. Under suitable smoothness assumptions, we obtain an absolutely convergent triadic expansion and quantitative bounds that distinguish local and nonlocal contributions in scale space. This framework provides a transparent and fully explicit description of how energy transfer is mediated by triadic interactions and how scale locality emerges as a structural property of the nonlinearity. Building on this formulation, we revisit the classical inertial-range picture of turbulence from a deterministic perspective. We show that, under a scale-invariant flux assumption, the Kolmogorov scaling is formally consistent with the triadic energy-transfer mechanism at a structural level. The result does not rely on statistical assumptions, but instead follows from the structural properties of the Navier–Stokes equations combined with a scale-resolved representation of the energy flux. The present work thus provides a coherent synthesis of triadic interaction analysis, dyadic scale decomposition, and classical turbulence phenomenology, offering a deterministic framework that clarifies how Kolmogorov-type scaling constraints arise in the scale-resolved structure of the underlying equations.
More information:
Bertram, E., Physica D: Nonlinear Phenomena (Elsevier), 2026
A Momentum-Regulated Model For Star Formation Efficiency in Giant Molecular Clouds
We present a minimal analytic framework to investigate the star formation efficiency per free-fall time, $\epsilon_{\rm ff}$, in giant molecular clouds (GMCs), focusing on the origin of the observed clustering around $\epsilon_{\rm ff} \sim 0.01$. We modeled the time evolution of the turbulent velocity dispersion through a momentum balance between stellar feedback and turbulent dissipation, and we show that this generically leads to a stable low-efficiency equilibrium with only weak dependence on global cloud properties. We extended the framework by including a phenomenological contribution from gravity-driven turbulence and find that both feedback- and gravity-driven motions converge to similar equilibrium states under typical GMC conditions. The efficiency can be expressed as the ratio between a gravitational velocity scale and an effective feedback velocity scale, providing a physically transparent interpretation of self-regulated star formation. The model provides a simple, physically motivated interpretation of observed gas--star formation scaling relations, including a Schmidt-like scaling at cloud scales and a Kennicutt-like scaling when averaged over cloud populations. Comparison with observed GMC properties shows agreement within a factor of a few and highlights the weak sensitivity of $\epsilon_{\rm ff}$ to cloud parameters. Despite its simplicity, the framework captures the leading-order interplay between turbulence, gravity, and feedback, and it provides a physically transparent explanation for the origin and robustness of low star formation efficiencies in GMCs.
Figure: Predicted star formation efficiency per free-fall time, $\epsilon_{\rm ff}$, as a function of gas surface density, $\Sigma_{\rm gas}$, for a set of representative cloud models with varying sizes $L$ and momentum injection efficiencies $p_*/M_*$. The shaded orange region indicates the typical observational range, while the gray band marks the characteristic surface density range of Milky Way giant molecular clouds ($\Sigma \sim 50$--180 $M_\odot\,\mathrm{pc}^{-2}$). The normalization depends on the adopted value of $\eta$ (here we use $\eta = 5$), but the weak dependence on $\Sigma$ remains unchanged. The model predicts a weak increase of $\epsilon_{\rm ff}$ with surface density, but over the limited range occupied by typical GMCs the efficiency remains approximately constant at the $\sim 1\%$ level. Outside this regime, systematic deviations emerge, with lower efficiencies at low surface densities and higher efficiencies at large $\Sigma$, reflecting the limits of the momentum-regulated equilibrium in more diffuse or extreme environments. The black points with error bars show observed molecular clouds. The data lie within the region spanned by the model curves and cluster around the characteristic efficiency, with a scatter comparable to the observational uncertainties.
More information:
Bertram, E., Astronomy & Astrophysics, 2026
A Momentum-Regulated Model For Star Formation Efficiency in Giant Molecular Clouds
From customer survey feedback to software improvements: Leveraging the full potential of data
Converting customer survey feedback data into usable insights has always been a great challenge for large software enterprises. Despite the improvements on this field, a major obstacle often remains when drawing the right conclusions out of the data and channeling them into the software development process. In this paper we present a practical end-to-end approach of how to extract useful information out of a data set and leverage the information to drive change. We describe how to choose the right metrics to measure, gather appropriate feedback from customer end-users, analyze the data by leveraging methods from inferential statistics, make the data transparent, analyze large volumes of user comments efficiently with Large Language Models, and finally drive change with the results. Furthermore, we present an example of a UX dashboard that can be used to communicate the analyses to stakeholders within the company.
Figure: Another UX prototype dashboard showing the UEQ scores and their 95% confidence intervals for six products over time. It also shows the split between the products' pragmatic and hedonic qualities. Again, by combining different filter settings at the top one can try to extract additional information out of the data set.
More information:
Bertram, E., Hollender, N., Juhl, S., Loop, S., Schrepp, M., Springer Nature, 2024
From customer survey feedback to software improvements: Leveraging the full potential of data
Figure: Logarithmic maps of the CO-to-H2 conversion factor for the alpha = 2.0 model. We see that only the inner parts of the cloud yield the canonical X_CO-factor value on average, while the outer cloud regions exhibit significantly larger X_CO values. The gray background denotes regions where the CO emission is zero. Hence, it is impossible to compute a value for X_CO there.
More information:
Bertram, E., Glover, S. C. O., Clark, P. C., Ragan, S. E., Klessen, R. S., MNRAS, 2015
Synthetic observations of molecular clouds in a galactic center environment: I. Studying maps of column density and integrated intensity
Synthetic observations of molecular clouds in a galactic center environment: I. Studying maps of column density and integrated intensity
We run numerical simulations of molecular clouds (MCs), adopting properties similar to those found in the Central Molecular Zone (CMZ) of the Milky Way. For this, we employ the moving mesh code Arepo and perform simulations which account for a simplified treatment of time-dependent chemistry and the non-isothermal nature of gas and dust. We perform simulations using an initial density of n_0 = 10^3 cm^{-3} and a mass of 1.3x10^5 M_sun. Furthermore, we vary the virial parameter, defined as the ratio of kinetic and potential energy, alpha = E_{kin} / |E_{pot}|, by adjusting the velocity dispersion. We set it to alpha = 0.5, 2.0 and 8.0, in order to analyze the impact of the kinetic energy on our results. We account for the extreme conditions in the CMZ and increase both the interstellar radiation field (ISRF) and the cosmic-ray flux (CRF) by a factor of 1000 compared to the values found in the solar neighbourhood. We use the radiative transfer code RADMC-3D to compute synthetic images in various diagnostic lines. These are [CII] at 158 micron, [OI] (145 micron), [OI] (63 micron), 12CO (J = 1 -> 0) and 13CO (J = 1 -> 0) at 2600 micron and 2720 micron, respectively. When alpha is large, the turbulence disperses much of the gas in the cloud, reducing its mean density and allowing the ISRF to penetrate more deeply into the cloud's interior. This significantly alters the chemical composition of the cloud, leading to the dissociation of a significant amount of the molecular gas. On the other hand, when alpha is small, the cloud remains compact, allowing more of the molecular gas to survive. We show that in each case the atomic tracers accurately reflect most of the physical properties of both the H2 and the total gas of the cloud and that they provide a useful alternative to molecular lines when studying the ISM in the CMZ.
Star formation efficiencies of molecular clouds in a galactic center environment
We use the Arepo moving mesh code to simulate the evolution of molecular clouds exposed to a harsh environment similar to that found in the galactic center (GC), in an effort to understand why the star formation efficiency (SFE) of clouds in this environment is so small. Our simulations include a simplified treatment of time-dependent chemistry and account for the highly non-isothermal nature of the gas and the dust. We model clouds with a total mass of 1.3x10^5 M_{sun} and explore the effects of varying the mean cloud density and the virial parameter, alpha = E_{kin}/|E_{pot}|. We vary the latter from alpha = 0.5 to alpha = 8.0, and so many of the clouds that we simulate are gravitationally unbound.
We expose our model clouds to an interstellar radiation field (ISRF) and cosmic ray flux (CRF) that are both a factor of 1000 higher than the values found in the solar neighbourhood. As a reference, we also run simulations with local solar neighbourhood values of the ISRF and the CRF in order to better constrain the effects of the extreme conditions in the GC on the SFE. Despite the harsh environment and the large turbulent velocity dispersions adopted, we find that all of the simulated clouds form stars within less than a gravitational free-fall time. Increasing the virial parameter from alpha = 0.5 to alpha = 8.0 decreases the SFE by a factor ~4-10, while increasing the ISRF/CRF by a factor of 1000 decreases the SFE again by a factor ~2-6. However, even in our most unbound clouds, the SFE remains higher than that inferred for real GC clouds. We therefore conclude that high levels of turbulence and strong external heating are not enough by themselves to lead to a persistently low SFE at the center of the Galaxy.
Figure: Star formation efficiencies per free-fall time against the corresponding virial parameter for our fiducial density models. We generally find a decreasing trend of the SFEs with higher value of the virial parameter for all density models. Although there are slight differences in the individual SFEs between the two seed models, we nevertheless find the same general trends of decreasing SFE with increasing α. To illustrate the different trends, we also fit exponential functions to the models and list the corresponding slopes in the plot. Error bars indicate variations of the SFE of ∼ 30%.
More information:
Bertram, E., Glover, S. C. O., Clark, P. C., Klessen, R. S., MNRAS, 2015
Star formation efficiencies of molecular clouds in a galactic center environment
Structure analysis of simulated molecular clouds with the Delta-variance
We employ the Delta-variance analysis and study the turbulent gas dynamics of simulated molecular clouds (MCs). Our models account for a simplified treatment of time-dependent chemistry and the non-isothermal nature of the gas. We investigate simulations using three different initial mean number densities of n_0 = 30, 100 and 300 cm^{-3} that span the range of values typical for MCs in the solar neighbourhood. Furthermore, we model the CO line emission in a post-processing step using a radiative transfer code. We evaluate Delta-variance spectra for centroid velocity (CV) maps as well as for integrated intensity and column density maps for various chemical components: the total, H2 and 12CO number density and the integrated intensity of both the 12CO and 13CO (J = 1 -> 0) lines. The spectral slopes of the Delta-variance computed on the CV maps for the total and H2 number density are significantly steeper compared to the different CO tracers. We find slopes for the linewidth-size relation ranging from 0.4 to 0.7 for the total and H2 density models, while the slopes for the various CO tracers range from 0.2 to 0.4 and underestimate the values for the total and H2 density by a factor of 1.5-3.0. We demonstrate that optical depth effects can significantly alter the Delta-variance spectra. Furthermore, we report a critical density threshold of ~100 cm^{-3} at which the Delta-variance slopes of the various CO tracers change sign. We thus conclude that carbon monoxide traces the total cloud structure well only if the average cloud density lies above this limit.
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Figure: Delta-variance spectra for the turbulent velocity field of our H2 and CO density models for a fixed initial number density of n_0 = 100 cm^{-3} and a resolution of 512^3 grid cells, computed for the centroid velocity (CV) maps. In order to analyze the impact of the filter function and its diameter ratio on our results, we compute the H2 and CO density spectra using both a Mexican hat with a diameter ratio of 1.5 and a French hat with a diameter ratio of 3.0. We do not find any significant differences in the slopes within the fitting errors if we use a another filter function for the Delta-variance analysis combined with a different diameter ratio.
More information:
Bertram, E., Klessen, R. S., Glover, S. C. O., MNRAS, 2015
Structure analysis of simulated molecular clouds with the Delta-variance
Centroid Velocity Statistics of Molecular Clouds
We employ the Delta-variance analysis and study the turbulent gas dynamics of simulated molecular clouds (MCs). Our models account for a simplified treatment of time-dependent chemistry and the non-isothermal nature of the gas. We investigate simulations using three different initial mean number densities of n_0 = 30, 100 and 300 cm^{-3} that span the range of values typical for MCs in the solar neighbourhood. Furthermore, we model the CO line emission in a post-processing step using a radiative transfer code. We evaluate Delta-variance spectra for centroid velocity (CV) maps as well as for integrated intensity and column density maps for various chemical components: the total, H2 and 12CO number density and the integrated intensity of both the 12CO and 13CO (J = 1 -> 0) lines. The spectral slopes of the Delta-variance computed on the CV maps for the total and H2 number density are significantly steeper compared to the different CO tracers. We find slopes for the linewidth-size relation ranging from 0.4 to 0.7 for the total and H2 density models, while the slopes for the various CO tracers range from 0.2 to 0.4 and underestimate the values for the total and H2 density by a factor of 1.5-3.0. We demonstrate that optical depth effects can significantly alter the Delta-variance spectra. Furthermore, we report a critical density threshold of ~100 cm^{-3} at which the Delta-variance slopes of the various CO tracers change sign. We thus conclude that carbon monoxide traces the total cloud structure well only if the average cloud density lies above this limit.
Figure: Delta-variance spectra for the turbulent velocity field of our H2 and CO density models for a fixed initial number density of n_0 = 100 cm^{-3} and a resolution of 512^3 grid cells, computed for the centroid velocity (CV) maps. In order to analyze the impact of the filter function and its diameter ratio on our results, we compute the H2 and CO density spectra using both a Mexican hat with a diameter ratio of 1.5 and a French hat with a diameter ratio of 3.0. We do not find any significant differences in the slopes within the fitting errors if we use a another filter function for the Delta-variance analysis combined with a different diameter ratio.
More information:
Bertram, E., Klessen, R. S., Glover, S. C. O., MNRAS, 2015
Structure analysis of simulated molecular clouds with the Delta-variance
Principal Component Analysis of Molecular Clouds: Can CO reveal the dynamics?
We use Principal Component Analysis (PCA) to study the gas dynamics in numerical simulations of typical MCs. Our simulations account for the non-isothermal nature of the gas and include a simplified treatment of the time-dependent gas chemistry. We model the CO line emission in a post-processing step using a 3D radiative transfer code. We consider mean number densities n_0 = 30, 100, 300 cm^{-3} that span the range of values typical for MCs in the solar neighbourhood and investigate the slope \alpha_{PCA} of the pseudo structure function computed by PCA for several components: the total density, H2 density, 12CO density, 12CO J = 1 -> 0 intensity and 13CO J = 1 -> 0 intensity. We estimate power-law indices \alpha_{PCA} for different chemical species that range from 0.5 to 0.9, in good agreement with observations, and demonstrate that optical depth effects can influence the PCA. We show that when the PCA succeeds, the combination of chemical inhomogeneity and radiative transfer effects can influence the observed PCA slopes by as much as ~ +/- 0.1. The method can fail if the CO distribution is very intermittent, e.g. in low-density clouds where CO is confined to small fragments.
Figure: Integrated intensity maps for 12CO on a linear scale computed along the LoS in z-direction for different times and turbulent velocity fields for models n100 (top) and n300 (bottom), corresponding to runs with inital number densities of 100 and 300 cm^{-3}. In model n100 there is less CO emission produced at low densities than in the n300 model. Note the different scaling in the colorbars.
More information:
Bertram, E., Shetty, R., Glover, S. C. O., Klessen, R. S., Roman-Duval, J., Federrath, C., MNRAS, 2014
Principal Component Analysis of Molecular Clouds: Can CO reveal the dynamics?
Statistical analysis of the mass-to-flux ratio in turbulent cores: effects of magnetic field reversals and dynamo amplification
We study the mass-to-flux ratio (M/\Phi) of clumps and cores in simulations of supersonic, magnetohydrodynamical turbulence for different initial magnetic field strengths. We investigate whether the (M/\Phi)-ratio of core and envelope, R = (M/\Phi)_{core}/(M/\Phi)_{envelope} can be used to distinguish between theories of ambipolar diffusion and turbulence-regulated star formation. We analyse R for different Lines-of-Sight (LoS) in various sub-cubes of our simulation box. We find that, 1) the average and median values of |R| for different times and initial magnetic field strengths are typically greater, but close to unity, 2) the average and median values of |R| saturate at average values of |R| ~ 1 for smaller magnetic fields, 3) values of |R| < 1 for small magnetic fields in the envelope are caused by field reversals when turbulence twists the field lines such that field components in different directions average out. Finally, we propose two mechanisms for generating values |R| ~< 1 for the weak and strong magnetic field limit in the context of a turbulent model. First, in the weak field limit, the small-scale turbulent dynamo leads to a significantly increased flux in the core and we find |R| ~< 1. Second, in the strong field limit, field reversals in the envelope also lead to values |R| ~< 1. These reversals are less likely to occur in the core region where the velocity field is more coherent and the internal velocity dispersion is typically subsonic.
Figure: Logarithmic column density map computed along the LoS in y-direction for an initial plasma beta of 0.01 (i.e. for a very strong field, left). One can see the outstanding z-direction of the magnetic field lines. Also labelled are the positions of 40 density-peaks (dot in the middle of each circle), which fulfil a threshold condition and the maximum diameter of the envelopes (circles).
More information:
Bertram, E., Federrath, C., Banerjee, R., Klessen, R. S., MNRAS, 2011
Statistical analysis of the mass-to-flux ratio in turbulent cores: effects of magnetic field reversals and dynamo amplification