At 2:00pm, room E349, Pierre Auclair will be talking about

Excursion-set for Primordial Black Holes: white noise and moving barrier

I will introduce pedagogically the Excursion-Set formalism, a framework in which the mass distribution of primordial black holes (PBHs) is derived from the first-passage time of a random walk describing the density contrast as the coarse-graining scale varies. In particular, I will address in detail two recent criticisms/remarks that have been raised about this approach. First, it was argued that the random walks are subject to colored (i.e. correlated over time) noise, making the first-passage-time problem cumbersome. We show that this arises from an incorrect separation of drift and noise when sampling on the Hubble-crossing surface: if Fourier modes are uncorrelated, the noise is strictly white. Moreover, sampling along the Hubble-crossing surface precludes using the density dispersion as a time variable, explaining the reported pathologies. Sampling instead on a synchronous surface removes both issues. This requires solving a first-passage-time problem with a moving barrier, for which we provide an efficient numerical framework. Second, it was suggested that cloud-in-cloud (i.e. that large black holes may engulf smaller ones) is irrelevant for PBHs and that the excursion set is therefore not needed. While valid for widely separated scales, this statement fails for broad power spectra with enhanced continua of modes. We further show that Press-Schechter estimates neglecting boundary evolution can break down even without cloud-in-cloud effects [1].

There are reasons to believe the early Universe was dominated by a scalar field which caused space to expand in an accelerating manner. After this Cosmic Inflation, the scalar field was replaced by radiation consisting of Standard Model particles. The subsequent radiation-dominated era is well understood. For example, the nuclei of light elements were born during it through the process of Big Bang Nucleosynthesis (BBN). Theoretical computations of BBN match very well the observed late-time abundances of these elements (with the possible exception of lithium). Hence, by BBN, the Universe must have settled to the standard radiation-dominated behaviour, and any exotic ingredients beyond the Standard Model must have vanished or become negligible. However, the era between inflation and BBN is less constrained, and exotic processes may have taken place there.

It is, in fact, possible that scalar fields may have dominated the Universe for some time even after the end of inflation. This happens in models of quintessential inflation, where the scalar field responsible for inflation sticks around and later comes to play the role of dark energy, causing the Universe to inflate again at late times. Right after the primordial inflation, the scalar field typically undergoes kination, motion dominated by the scalar’s kinetic energy. A long period of kination leaves a mark: it amplifies gravitational waves born during inflation. This has a two-fold effect. First, the amplified gravitational waves may become detectable in future gravitational wave observatories. Second, the gravitational waves may break the delicate balance of BBN, producing wrong ratios of light elements and making the model unviable.

The second effect tends to kick in before the first: observationally interesting models are ruined by their inability to abide by the BBN bound. In Ref. [1], Eemeli worked together with Konstantinos Dimopoulos and Lucy Brissenden from Lancaster University to show that there is a class of kination models that respects the BBN bound but is still detectable by future gravitational wave experiments. The trick is to make the transition from inflation kination (called a ‘stiff’ period) ‘soft,’ so that the Universe’s equation of state changes slowly during this era. The effect on the gravitational wave spectrum is depicted in the following figure:

The colored curves represent the sensitivities of current and future gravitational wave (and other) experiments for various gravitational wave frequencies. BBN sets a universal limit on the gravitational wave spectrum across all frequencies. The semi-vertical lines are two example spectra, one with standard kination (dashed blue) and one with a soft transition period (purple). The standard case increases steeply all the way to the highest frequencies, narrowly avoiding the BBN bound, but also missing most of the gravitational wave detectors’ sensitivities. The soft case has a rounded top (corresponding to the soft period), again avoids the BBN, but this time also hits the sensitivities of the Einstein Telescope (ET) and the Cosmic Explorer (CE). The maximum frequency for both curves is approximately fixed by the energy scale of inflation, which is here set to the highest observationally allowed value, since this also produces the highest gravitational wave density.

The soft period in [1] was produced by a double-exponential scalar potential motivated by string theory. With only a little tuning, the model produces both detectable gravitational waves and the correct dark energy behaviour at late times. The model also satisfies other necessary constraints, such as the requirement for the scalar field itself to be sufficiently suppressed during BBN. It paves the way for other models that can produce similar effects.

As discussed in, for example, here, cosmic inflation is an early era during which a scalar field makes the Universe expand extremely fast. Small ripples in the field can get amplified, so that some patches of space end up with more matter than others when inflation ends. If the extra matter is compact enough, it collapses into a primordial black hole. Computing the black hole statistics from a given model of inflation is an active field of study.

Compactness of matter is measured with the compaction function \(\mathcal{C}\), the extra mass \(\Delta M\) in a given patch of space divided by the patch’s radius \(R\). To see if a black hole forms around a given point in space, we can compute the compaction function for many patches of varying radii around the point. These give the radial profile of the compaction function. The profile goes to zero for small and large radii (where \(\Delta M\) becomes small and \(R\) becomes large, respectively), but may have a high peak at intermediate scales. If the peak is high enough, it triggers black hole collapse. The radius at the peak gives the black hole’s mass.

The compaction function is tricky to compute from models of inflation; indeed, usually cosmologists like to work with another perturbation quantity, the curvature perturbation \(\zeta\). The two quantities are related but not identical: the compaction function is related to the curvature perturbation’s radial derivative. It is commonly assumed that a high curvature perturbation corresponds to a high compaction function, so that the curvature perturbation can be used to estimate the abundance of primordial black holes. In Ref. [1], together with Syksy Räsänen and Sami Raatikainen from the University of Helsinki, Eemeli has shown that this is not always the case.

In Ref. [1], the authors performed a numerical study of inflationary perturbations using the method of stochastic inflation (for more on stochastic inflation, see these posts: arXiv:2507.15522, arXiv:2504.17680, and arXiv:2501.05371. With Finnish supercomputers, they built \(10^8\) radial perturbation profiles in three typical single-field primordial black hole models. The study is the first one to construct the full radial profiles; previous studies have been limited to studying the perturbations only at fixed length scales. This was achieved through a Fourier transform of over 10 000 correlated momentum shells, revealing the full correlations between different radii. Interestingly, the compaction profiles turned out to be very spiky, with sharp dips up and down instead of wide, smooth peaks. The figure below shows an example case: on the left, the curvature perturbation profile, and on the right, the corresponding compaction function profile (together with various estimates; for details, see Ref. [1]).

Previous studies have mainly used the curvature perturbation amplitude as a criterion for black hole formation. The statistics of the radial profiles show, however, that the maximum amplitudes of the compaction function and the curvature perturbation are not very strongly correlated, see the figure below:

Previous work has argued that black hole abundance is enhanced by non-perturbative effects that modify the curvature perturbation distribution, giving it an exponential tail. The results of [1] indicate that the spikiness of the profiles has an even larger effect on the abundance, increasing it by many orders of magnitude compared to previous estimates. The compaction function spikes rise well over typical mean profiles deduced from the curvature perturbation amplitude, forming black holes even in cases where the curvature perturbation stays small. As a consequence, the curvature perturbation spectrum required to form abundant black holes is smaller than previously assumed, of order 0.001 by a preliminary estimate. Since the spikes can form at many different scales, the black hole mass distribution becomes wide, as shown below. Such a wide distribution may clash with observational constraints.

The results of [1] come with caveats. The collapse threshold used in this work is based on earlier numerical relativity simulations, which consider gravitational collapse starting from smooth profiles. The narrow and spiky compaction function peaks found here are quite different, and new collapse simulations are needed to see how this affects the collapse threshold. The wide mass distributions also suffer from convergence issues, and it is possible that the large-scale peaks get wiped out by cosmological evolution before they have time to collapse into black holes. More work is needed extract all information from the spiky perturbation profiles.

At 2:00pm, room E349, Tomo Takahashi will be talking about

More fields are different: Stochastic view of multi-field inflationary scenario

High-energy physics often motivates multi-field inflationary scenarios where stochastic effects play a crucial role. Peculiar to multi-field models, the noise-induced centrifugal force results in a longer duration of inflation depending on the number of fields, even when the stochastic noises themselves are small. We show that, in such small-noise regimes, the number of fields generically discriminates whether inflation successfully terminates or lasts forever. Our results indicate that inflation with an extremely large number of fields may fail to realise our observable Universe.

At 2:00pm, room E349, Lisa Mickel will be talking about

Trajectories for a superposition quantum universe

In the quest to establish possible imprints of quantum gravity one can quantise the reduced FLRW phase space to obtain so-called minisuperspace models that should be understood as a low energy limit of a full quantum gravitational theory. For a minisuperspace model that uses a perfect fluid as internal time and an affine quantisation procedure one finds solutions for the wave function that correspond to bouncing universes, thus resolving the big bang singularity. Starting from such a bouncing minisuperspace model we will relax the assumption that the state of the quantum universe has to correspond to a single highly semiclassical state. Instead, we will consider a universe in a superposition of such states, in accordance with the superposition principle of quantum mechanics. We are then faced with the question of how to extract a semiclassical scale factor that can be connected to the classical evolution of GR from such a system. Here we propose to make use of the trajectory approach to quantum mechanics, which assigns a unique value to the scale factor at all times. We introduce this approach to quantum mechanics more generally and show how it can be used to describe a single state universe, before continuing to investigate trajectories obtained for the case of a biverse (superposition of two semiclassical states). These trajectories exhibit features that significantly differ from the single state case. Finally, we illustrate the treatment of tensor perturbations on a background given by a biverse trajectory and consider the implications of the existence of such a trajectory for the evolution of perturbations.

At 2:00pm, room E349, José Beltrán Jiménez will be talking about

Electromagnetically K-mouflaged dark matter and the ladder symmetries of screened objects

I will discuss a cosmological scenario where dark matter is provided with a dark electric charge with a dark electromagnetic sector featuring a screening mechanism so that all the effects only appear at low redshift, when dark matter is sufficiently clustered. Within these models, it is natural to have a universe described by a Lemaitre model instead of a FLRW. Instead of solving the full relativistic equations, I will consider a Newtonian approach that is sufficient for a matter dominated universe with, possibly, a cosmological constant. In this scenario, it is possible to explain the Hubble tension in terms of the dark electric repulsion between dark matter halos. After reviewing some phenomenological consequences, I will proceed to analysing the deformability of dark matter haloes due to the dark electromagnetic interaction and show the emergence of a curious 2-ladder structure connecting different multipoles for the case of Born-Infeld electromagnetism. Furthermore, some multipoles exhibit a vanishing polarisability or magnetisation, thus showing an intriguing resemblance with the vanishing of the Love numbers of black holes. Finally, I will explain how analogous results arise for the scalar DBI theories in arbitrary dimensions and some finite size effects.

At 11:00am, room E349, José Beltrán Jiménez will be talking about

The Cosmological Principle: its non-trivial realisations with some phenomenological consequences

The Cosmological Principle is one of the pillars of the standard model of cosmology and it is commonly realised in a trivial way with homogeneous SO(3)-scalars. I will discuss several scenarios where the matter sector realises the Cosmological Principle in a non-trivial manner by resorting to combinations of spacetime and internal symmetries. These scenarios include the effective field theory of fluids and solids (as well as their dual formulations), but more general setups based on e.g. vector fields with internal symmetries. A natural consequence of some of these scenarios is the appearance of a second helicity-2 mode in the cosmological perturbations that produces oscillations of gravitational waves with distinctive signatures. I will also discuss some intriguing realisations that occur on shell and lead to the possibility of having preferred directions in an exactly isotropic background universe.

Cosmic inflation is a period of accelerated expansion of space which likely took place in the very early universe. The expansion is driven by the inflaton, a scalar field that undergoes random motion due to quantum effects. This random motion shows up as inhomogeneities in the late universe, seeding cosmic structure such as galaxies and galaxy groups. If the inhomogeneities are particularly strong at short scales, they may form structures even before galaxy formation, collapsing straight into primordial black holes. Such black holes would be dark matter and would radiate gravitational waves that could be observable by gravitational wave detectors.

A popular way to produce the strong inhomogeneities needed for black hole formation is to give the inflaton an inflection point potential with a local bump (see figure). As the inflaton climbs over the bump, it slows down, so the quantum fluctuations can have a stronger relative effect. Besides black holes, such strong quantum effects may lead to another phenomenon: eternal inflation. In eternal inflation, quantum kicks oppose classical forces enough to keep the inflaton near the bump eternally in some regions of space.

To test if inflation is eternal in primordial black hole models, in Ref. [1], we solved the Fokker-Planck equation, which describes the time evolution of the inflaton’s probability distribution (\(N\) is the time variable):

The classical drift \(\mu\) and the quantum diffusion coefficient \(\sigma\) depend on the model, but take universal forms near the potential’s local minimum and maximum, which are most important for eternal inflation. We were able to show that eternal inflation happens generically near the minimum if field variations are Planckian, \(\phi \sim M_\text{Pl}\), and also near the maximum if the potential’s second derivative there is not too large in magnitude, \(|V''(\phi)| \leq 6V(\phi)/M_\text{Pl}^2\). We checked these conditions against three typical models from the literature, and they were satisfied in all three cases. Eternal inflation is a generic consequence of such inflection point models.

Why is eternal inflation interesting? Eternal inflation fractures space into a multiverse of different regions, each of which exits the eternally inflating bulk at a different time. Making cosmological predictions then becomes complicated: different regions have undergone different inflationary histories which may show up as different structures and inhomogeneities in the late universe. This is sketched in the figure below: the “usual” primordial black hole space time is marked as U1, but where the inhomogeneities are large - i.e. inside the black holes - there are regions which continue inflation eternally (E) or exit in a couple of different fashions (U2, U3). The “new universes” U1 and U2 still see the edge of the E region, which makes them violently inhomogeneous at large scales, incompatible with the cosmos we see around us.

Since the volume of the eternally inflating universe is infinite, it is not easy to assign probabilities for the different universes U1, U2, and U3. This is called the measure problem. However, in our case, all reasonable solutions to the measure problem seem to suggest that the U1 universe (the one that could match our observable universe) is the rarest one, since the others “branch out” of it in an infinite fashion. In other words, eternal inflation presents a challenge to primordial black hole models: the homogeneous pockets of space with black holes are actually the last places we would expect to find ourselves in. To solve this challenge, we may need to consider new types of primordial black hole models or rethink our approach to the measure problem.

At 2:00pm, room E349, Carlos J.A.P. Martins will be talking about

To scale, or not to scale

Cosmic defect networks are fossil relics of earlier stages in the universe’s evolution. Their astrophysical detection would provide unique insights into these earlier stages, and even upper limits yield valuable information. However, for these analyses to be reliable, one needs a precise and accurate understanding of the evolution of these networks. I will describe recent progress, both on analytic modelling and on numerical simulations, to improve the understanding of this evolution. The focus will be on the issue of scaling of the networks. The linear scaling solution is known to be an attractor for the simplest defects, such as plain cosmic strings, but in more general (and arguably more realistic) models, such as those in which the defects have worldsheet charges and currents, other cosmological scaling solutions can in principle occur, depending on the microphysics of the model. I will present a taxonomy of these solutions, and briefly comment on how different models could be distinguished in the event of future detections.

Cosmic Inflation is the currently favoured scenario of the early universe and refers to an accelerated expansion of the spacetime in its earliest instants. See this post for state-of-the-art observational constraints.

Among the simplest inflationary scenarios, the so-called plateau models may exhibit a very flat potential, as it is the case for the Starobinsky model plotted below. Cosmic inflation occurs in the semi-classical regime and cosmological perturbations are generated for field values around \(\phi_*\) (see figure).

However, there is also a field value, \(\phi_{\mathrm{qw}}\), above which quantum fluctuations dominate, here referred to as the “quantum domain”. The field evolution in there is stochastic, \(\phi\) can move up or down the potential while the universe keeps inflating. The quantum fluctuations being driving the dynamics also implies that the spacetime can become strongly curved, in ways which cannot be easily calculated with semi-classical methods.

Putting numbers together, one finds \(\phi_* \simeq 5 \, M_{\mathrm{Pl}}\) and \(\phi_{\mathrm{qw}} \simeq 47 \,M_{\mathrm{Pl}}\). In other words, the quantum domain is very far away from everything than can be observed in Cosmology. In fact, we can associate some length scales to this region by counting how much semi-classical expansion occurs between \(\phi_\mathrm{qw}\) and \(\phi_*\). We find that the quantum domain may affect our universe on length scales larger than \(10^{10^{16}}\,\mathrm{Gpc}\), a tetration. Therefore, cosmic inflation extended with stochastic inflation implies that quantum physics is in charge not only within the microscopic world but also on the largest possible distances. Why bothering? First, as we discuss in this post, some residual gradients of these inhomogeneities might create a non-vanishing curvature in our observable Hubble volume, and this could be a mean to determine, given some curvature measurement, how likely stochastic inflation is. Then, on more fundamental grounds, this quantum domain is a natural extension of what we expect from cosmic inflation in plateau potentials, understanding the structure of the universe it is creating is what Cosmology is all about.

Working out the dynamics of the field-metric system when \(\phi > \phi_\mathrm{qw}\) can be made using the so-called stochastic inflation formalism (see this post for an application to primordial black holes). It is an effective field theory in which quantum fluctuations on sub-Hubble scales are integrated out to determine the field and metric evolution on super-Hubble scales, the latter emerging as real stochastic variables.

Unfortunately, when applied to a potential \(V(\phi) = 3 H_{\mathrm{inf}}^2\) being exactly constant for \(\phi > \phi_\mathrm{qw}\), stochastic inflation leads to divergences and that is preventing the determination of the spacetime curvature. Indeed, assuming that stochastic inflation started at \(\phi_0\), at a time \(N_0\), it stops when the quantum fluctuations push the field away from the quantum domain, i.e. when \(\phi = \phi_{\mathrm{qw}}\) at a time \(N_{\mathrm{qw}}\). Using the stochastic formalism, one can show that the mean value of the lifetime

In other words, stochastic inflation lasts, on average, an infinite amount of time. However, we are necessarily living in one of the realizations in which stochastic inflation ended. As such, when concerned with cosmological observables, we should only be concerned with the set of all quantum realizations in which stochastic inflation ends.

A way to enforce the ending condition is to reverse the flow of time: the quantum field starts at \((\phi_{\mathrm{qw}},N_{\mathrm{qw}})\) and evolves backward in times to end its evolution at \((\phi_0,N_0)\). Even if the idea is very simple, its practical implementation requires to mathematically perform what is called a time-reversion in diffusion processes. This is what we have done in Ref. [1] for stochastic inflation by reversing from the lifetime. Defining the reverse time by

together with \(\Delta \phi \equiv \phi - \phi_{\mathrm{qw}}\), the field value in reference to the quantum wall, the following picture shows one of these reverse-time realization.

This trick allows us to interpret stochastic inflation as a collection of random processes conditioned by their lifetimes \(\Delta N_0\). On the mathematical side, the associated probability distribution \(\bar{P}(\phi,\Delta N|\phi_0,\Delta N_0)\) has been shown to follow a reverse Fokker-Planck equation that we have solved, one of the solution being represented below. The colors encode how probable it is to find the field at a given value, at a given reverse time, and for a given lifetime \(\Delta N_0\) of the processes.

In the reverse-time stochastic inflation formalism, all divergences are gone and we were able to determine the distribution \(P(\zeta|\phi_0)\) of the spacetime curvature fluctuations \(\zeta\). It is a well-defined and normalizable distribution, which has been obtained by marginalization over all possible lifetimes, including the infinite limit. We find that it depends on one parameter only, the initial field excursion value \(\Delta \phi_0 = \phi_0-\phi_{\mathrm{qw}}\) expressed in unit of the Hubble parameter during inflation, i.e. the quantity

The probability distribution for \(\zeta\) is plotted below as a blue curve.

It is skewed towards positive values and has tails slowly decaying as \(|\zeta|^{-3/2}\). The orange curve is an analytical approximation matching very well the exact distribution in the tails. It reads

In qualitative terms, our result shows that nothing really bad occurs when stochastic inflation occurs within the flat semi-infinite potential. The distribution of \(\zeta\) ends up being peaked around vanishing values with a width typically given by \(\chi_0^2\). As such, it is very well possible that our observable universe is a classical slice sandwiched between two quantum worlds, the very small and the very large!