In inflationary cosmology, cosmic structures like galaxies find their origin in the quantum fluctuations occurring in the first instants of the universe history. Large and small structures today, represented in the following figure by two length scales \(\lambda_1\) and \(\lambda_2\), do not exit the Hubble radius during inflation at the same times. This time-shift at Hubble exit, combined with a small change of the Hubble radius itself, explains why cosmic structures are aggregated slightly differently at large and small distances. In technical terms, the power spectrum of the cosmological fluctuations is not scale invariant but has a spectral index which is, according to current observations, slightly red \(n_\mathrm{S} \lesssim 1\).

The condition \(n_\mathrm{S} \lesssim 1\) is a “natural” expectation
of the single-field scenarios depicted in the previous
picture. Indeed, the accelerated slow rolling of the field along its
potential necessarily triggers a small growth of the Hubble radius
during inflation. Other inflation models do exist in which the slow
rolling is actually decelerated and their predicted spectral index
ends up being blue (\(n_\mathrm{S} \gtrsim 1\)). These models are
however strongly disfavoured since the first measurements of
\(n_\mathrm{S}\) by the
**WMAP**
satellite (see also this **post**).

As can be seen in the previous figure, not only the Hubble radius grows but it also accelerates towards the end of inflation. This acceleration implies that the spectral index itself cannot be scale invariant, it should vary with length scales, and that is the definition of the running \(\alpha_\mathrm{S}\). As such, if our understanding of inflation is correct, measuring a red spectral index \(n_\mathrm{S} < 1\) necessarily implies a negative running \(\alpha_\mathrm{S} < 0\).

In Ref. [1], **Christophe**,
**Jérôme Martin** and **Vincent
Vennin** have
computed the marginalized posterior probability of the running of the
spectral index \(\alpha_\mathrm{S}\) given the latest Cosmic Microwave
Background and Baryon Acoustic Oscillations measurements (see this
**post** for more details on
the cosmological data used). The probability distribution is
represented below, along with various credible intervals, showing that
single-field inflationary models predict a negative running.

In numbers, we find (confidence given in parenthesis):

\[-8.8 \times 10^{-4} < \alpha_\mathrm{S} < -4.7 \times 10^{-4} \quad (68\%),\] \[-1.4 \times 10^{-3} < \alpha_\mathrm{S} < -2.4 \times 10^{-4} \quad (95\%),\] \[-1.8 \times 10^{-3} < \alpha_\mathrm{S} < -9.1 \times 10^{-5} \quad (98\%).\]These credible intervals are actual predictions and result from the current data being analysed under the theoretical prior that inflation is in the landscape of the slow-roll single-field models. As such, they are highly non-trivial.

For instance, dropping information from data, or, from the theoretical prior, would let us without any determination of the sign of \(\alpha_\mathrm{S}\). The following figure shows the posterior obtained by not assuming any specific inflationary scenario (“just slow-roll”) as a dashed blue curve. The posterior plotted earlier is reported here as a solid red curve (labelled “model space”). Allowed values of \(\alpha_\mathrm{S}\) under the “just slow-roll” hypothesis are orders of magnitude larger and the sign can be positive or negative.

If, instead, we drop the data while keeping the theoretical prior to be single-field models, one gets

The credible intervals are reported and they show that positive values of the running are allowed within the prior. Therefore, only the combination of the current cosmological data and the hypothesis that inflation occurred within the landscape of single-field models allows us to make the prediction that the running is negative.

]]>Cosmic Inflation is the leading explanation for the origin of cosmic
structures: these are seeded by quantum fluctuations occurring around
the event horizon of a exponentially fast accelerating space-time, see
this **post**. By measuring
the distribution of galaxies in our universe, the
**Euclid**
satellite is expected to provide very soon new exquisite measurements of these
fluctuations. Testing cosmic inflation will therefore require to have
exquisite predictions as well.

Observable predictions currently rely on the slow-roll approximation to determine the so-called e-fold times \(\Delta N=N−N_\mathrm{end}\), in reference to the e-fold \(N_\mathrm{end}\) at which inflation ended. These instants are actually used to map structures in the sky to quantum fluctuations during inflation. The precision at which they are determined is not so good, they are typically known at \(\mathcal{O}(1)\) e-fold precision. The following figure uses a full numerical integration of various inflationary models to compute the error made (vertical axis) using the slow-roll approximation as a function of the exact timing (horizontal axis).

In Ref. [1], we propose a new and simple
**velocity correction**, on top of slow-roll, that increases by one
order of magnitude the precision on \(\Delta N\). As
shown in the following figure, when compared to the exact solution,
our new method reaches an precision of about a tenth of e-fold (**blue
curve** compared to the red one).

The other curves (green and purple) show other corrections, improving the determination of \(\phi_\mathrm{end}\), the field value at which inflation ends. These ones may, or may not, improve over the velocity correction, depending on the model at hand.

]]>In its simplest incarnation, Cosmic Inflation can be realised by a
single field rolling down a potential and hundred of different
theoretical embeddings have been proposed since the advent of the
paradigm. Comparing these models to data could be viewed as Herculean
tasks and we have been sharing these enjoyments with **Jérôme
Martin** and **Vincent
Vennin** in the
past decade.

In Ref. [1], we present the most recent and
strongest constraints on single-field inflation imposed by the latest
cosmological data, namely the **Planck
satellite** 2020
Cosmic Microwave Background data, the **BICEP/Keck
array** 2021 polarization measurements, the
**South Pole Telescope**
third generation measurements and the full compilation of **Baryonic
Acoustic
Oscillations**
for the **Sloan Digital Sky
Survey**.

Our analysis incorporates the most accurate theoretical predictions
for the quantum fluctuations generated during inflation, the so-called
third-order slow-roll spectra for both primordial gravitational waves
and curvature fluctuations (see this **post**). The Bayesian model comparison analysis
uses some machine learning tools implementing the methods developed
in Ref. [2] and include all the new models
presented in the **Opiparous Edition of the Encyclopædia
Inflationaris**.

The following figure shows the one-dimensional marginalised posterior distributions obtained for the inflationary cosmological parameters, i.e., when the cosmic structures are seeded by the inflationary quantum fluctuations.

Notice that the third slow-roll parameter \(\epsilon_3\) is now constrained in a range perfectly consistent with slow-roll and we find

\(-0.44 < \epsilon_3 < 0.55 \quad (95\%\,\texttt{CL})\).

One may also notice the maximum probability for \(\epsilon_1\) at non-vanishing values, showing a weak statistical preference for the presence of primordial gravitational waves in the current data (mostly driven by the BICEP/Keck data). We have

\[\log(\epsilon_1) > 4.9 \quad (95\%\,\texttt{CL}),\]with

\[\log(\epsilon_1) < -2.6 \quad (98\%\,\texttt{CL}).\]The Bayes’ factors and maximum likelihood ratios for all models of the Encyclopædia Inflationaris are:

The reference model is denoted as “SR3” and represents the pure slow-roll analysis assuming no specific potential, only the natural priors for the slow-roll parameters \(\epsilon_i\in[-0.2,0.2]\). Bars extended to the left mean the models are models, they are favoured when the bar is extended to the right (with respect to agnostic slow-roll). The bottom labels give the Jeffreys’ scale of Bayesian evidence with respect to the best model. We find that \(40\%\) of all scenarios can be considered ruled-out (strongly disfavoured according to the Jeffreys’ scale) whereas \(20\%\) of the models are most probable given the current data.

Our approach also allows us to constrain the reheating epoch, the transition period between cosmic inflation and the hot Big-Bang phase in which the universe is a relativistic plasma.

The following figures are scattered plots of inflationary models positioned according to their Bayesian evidence (horizontal axis) and the information gain on the reheating epoch (vertical axis). The colour scale traces the mean value (over its posterior) of the reheating parameter \(\ln R_\mathrm{reh}\).

Each model is also encircled in a gauge counting the number of unconstrained model parameters (derived using Bayesian dimensionality). Non-encircled models are models for which all parameters are constrained by the data, models with a full circle around have all their parameters unconstrained (which are then superfluous to fit the data). As such, the most probable and most efficient models are those on the right having no circle around.

Weighted over the landscape, we find that the current data constrain the kinematics of reheating by \(1.3\) bits. The precision reached by the current cosmological data is such that almost half of the inflationary landscape is out of the game, but, also, for each of the favoured models, the way the universe reheated to become a plasma can now be inferred by astrophysical and cosmological observations. We are talking here of an epoch of the universe being, at least, at redshift \(z > 10^{10}\)!

The future is bright and we are looking forward to the **Euclid**
satellite measurements (see also this **post**).

At 2pm, room E349, **Eemeli Tomberg** (Lancaster University), will be talking about

*Quantum fluctuations from cosmic inflation give rise to the
macroscopic structures of the universe. The strongest fluctuations
collapse into primordial black holes, a dark matter candidate and a
possible source of gravitational waves. Stochastic inflation is a
tool to compute the fluctuation statistics non-perturbatively, needed
for accurate black hole predictions. I discuss recent progress in
these computations, their numerical implementation and analytical
approximations, and the implications for black hole abundance in
single-field models of inflation.*

**Hopes of Big Bang Discoveries Ride on a Future Spacecraft**

More details can be found in the scientific paper Ref. [1] and this **post**.

Primordial black holes (PBH) are cosmological objects that could have been formed in the earliest instants of the universe’s history, for instance, when large overdensities collapse under their own gravity. Understanding how these PBHs cluster is essential as it can provide insights into the evolution of structure formation in the universe and could potentially help in assessing how much, and when, binaries of these objects appear.

In Ref. [1], **Pierre**
and **Baptiste** have used the excursion-set
formalism to quantify the initial spatial clustering of PBHs generated
from large Gaussian density fluctuations. Their method takes into
account the “cloud-in-cloud” mechanism, which is a critical aspect
that is overlooked in studies using the Poisson model of clustering.

In the excursion-set formalism, the collapse of an overdensity to form a PBH of mass \(M\) is recast into the probability of finding the first crossing of a random walk at some scale \(S(M)\). This approach has been already applied to study the formation of large-scale structure, in which multiple crossings of a random walk describe a hierarchy of structures. This formalism was first developed to solve the “cloud-in-cloud” problem, that is resolving this hierarchy of structures.

The cloud-in-cloud mechanism implies that PBHs should be part of larger structures: they don’t just appear randomly as per the Poisson assumption. This results in a natural and intrinsic correlation between the formation of pairs of PBHs, which is important for accurately modelling the clustering behaviour at small scales. The following picture shows two random walks that share a common past until the “time” \(S_{r}\): they come from the same realisation of a large scale density perturbation. Subsequently, they evolve in distinct ways and this results in two different collapses that occur respectively at the first-crossing times \(S_{1}\) and \(S_{2}\)

Such an approach automatically includes short-range exclusion effects: PBHs are anti-correlated at short distances!

The authors also present explicit expressions for the excess probability to find pairs of PBHs separated by a given distance and for the excess probability to find pairs with an asymmetric mass ratio.

]]>LISA will be the next-generation observatory in the field of
gravitational waves, surpassing current terrestrial capabilities by
exploring lower frequencies and longer timescales, opening up a new
observation window into the universe. and technological
innovation. The LISA definition study report, in which **Pierre** was a
contributor, presents an in-depth analysis of the mission’s scientific
rationale, design, and implementation plans, see Ref. [1]. It highlights the
groundbreaking discoveries that LISA is expected to make

With ESA’s approval, the LISA mission moves one step closer to reality!

]]>Applicants interested in early universe cosmology (cosmic inflation,
reheating, cosmic defects…), CMB, large scale structures, gravitational
waves cosmology, as well as data analysis for current and future
cosmological observations (CMB-S4, Euclid, LiteBird, LISA…), are
particularly encouraged to apply. A summary of our **research
interests** can be consulted
online.

Appointment is available for two years. Please notice that applicants should have obtained their PhD no longer than 5 years before the starting date of the fellowship, plus one year of allowance for parenthood.

Letters of application (including a curriculum vitae, a list of publications, a brief statement of research interests) and at least two recommendation letters from senior scientists should be submitted on-line, by the 28th of January 2024, at:

**https://cp3.irmp.ucl.ac.be/job/94**

For more information, or postal applications, please contact:

```
christophe.ringeval@uclouvain.be
Cosmology, Universe and Relativity at Louvain
Institute of Mathematics and Physics
Louvain University
2, Chemin du Cyclotron
1348 Louvain-la-Neuve
Belgium
```

]]>