VIEWPOINT

Structure Formation in the Very EarlyUniverseNumerical calculations explain how density fluctuations in the Universe grew by orders ofmagnitude during the ‘‘primordial dark ages.’’

by Mark P. Hertzberg∗

A fundamental task of modern cosmology is to un-derstand the formation of structure in our Uni-verse from its earliest moments. The leadingidea is that of inflation [1, 2], which can suc-

cessfully explain many observed features of our Universe,from its flatness to its isotropy. Inflation produces an ex-

Figure 1: 3D rendering of density fluctuations calculated for auniverse that has expanded 200-fold since the end of inflation(brighter colors indicate larger densities). Musoke et al. find thatsmall-amplitude fluctuations at the end of inflation are amplified byseveral orders of magnitude. (N. Musoke et al., Phys. Rev. Lett.(2020))

∗Institute of Cosmology, Tufts University, Medford, MA, USA

tremely homogeneous Universe, but quantum fluctuationscreate fluctuations in density from place to place. Thesefluctuations become the seeds of the large-scale structure oftoday’s Universe—a foam-like texture of voids and galaxyclusters, sheets, and filaments. However, it is hard to ex-plain how the initial fluctuations grew in amplitude bymany orders of magnitude, leading to the present den-sity variations. Nathan Musoke and colleagues from theUniversity of Auckland, New Zealand, take small initialquantum fluctuations and predict their evolution with ahigh-precision numerical computation [3]. By focusing onlyon a specific regime—very early times and very small spatialscales—they are able to track the evolution of these fluctu-ations within a self-consistent computation. They predictthe development of complicated structures with density con-trasts that are orders-of-magnitude larger than the initialfluctuations. The results shed light on structure forma-tion in the first stages after inflation. They could also helpresearchers pinpoint new observational signatures of theearliest moments of the Universe.

The basic assumption of inflation is that the very earlyUniverse was filled with bosonic particles in a degeneratequantum state. These spinless particles, known as inflatons,are the quanta of a primordial field carrying a large potentialenergy that drove an exponential expansion of the Universe.Researchers have shown that, if they include the inflatonfield in Einstein’s field equations, they can predict a phaseof rapid exponential expansion, in which the Universe grewin size by over 30 orders of magnitude.

This inflationary expansion would lead to a universeteeming with inflaton particles distributed with extremelyuniform density. Describing the evolution of the inflatondistribution during the exponentially expanding phase ismoderately straightforward to do, at least for the simplestmodels for inflation. The inhomogeneities deriving frominitial quantum fluctuations are very small—deviations inlocal density are of order of 1 part in 100,000. In this regime,the corresponding Einstein equations can be simplified to ac-count only for linear perturbation terms—a readily tractablecase.

However, understanding what happens right after in-flation is much more difficult. In this postinflationary

physics.aps.org c© 2020 American Physical Society 10 February 2020 Physics 13, 16

era—dubbed the primordial dark ages because of the ab-sence of any photons—the inflaton dominates the Universe’senergy density, but all sorts of nonlinear effects can quicklybecome important and amplify inhomogeneities. The infla-tons can interact with each other, causing them to aggregateinto spatially distinct patches. Inflatons can also decay intoand couple to standard model particles. And gravitationalinteractions between all these various particles could be-come relevant [4].

To gain some insight into this era, Musoke et al. followan approach explored by previous work [5–8], which fo-cused on only one of those nonlinear factors: the interactionsof inflatons among themselves. Numerical studies of thispostinflationary era have recently been performed, but thecurrent work goes a step further by improving the numeri-cal code to track as accurately as possible the evolution fromthe linear to nonlinear regime.

Musoke et al. study the evolution of inflaton distributionby numerically solving with high precision the correspond-ing field equations coupled to gravity. This leads to a phasein which inflatons break up into a highly inhomogeneousconfiguration, and the system enters a nonrelativistic regimeas the particles’ wavelengths stretch and their momentadecrease. The researchers call this the matter-dominatedphase, as its evolution mirrors that which would be ob-served with conventional matter. However, since the systemis still highly degenerate, it cannot be described by classi-cal particle physics but, instead, by classical field theory. Todescribe these conditions, Musoke et al. use the so-calledSchrödinger-Poisson system of equations, which is knownto be numerically tractable [9]. The team relied on a numeri-cal code called PyUltraLight. Partly developed by them, thecode allows for a more accurate numerical treatment of theSchrödinger-Poisson equations.

In the matter-dominated era, density fluctuations growwith time. This is because gravity is attractive, so over-dense regions attract their surroundings and become evendenser. Starting from very small initial fluctuations Musokeet al. observe that fluctuations get amplified by several or-ders of magnitude, reaching or exceeding unity on somespatial scales (Fig. 1). This regime is precisely where lin-ear approximations fail. Thanks to their high-precision 3Dnumerical calculations, the team can tackle this regime andobtain statistical details of the resulting inflaton distribution.

The new work focuses only on inflatons and their self-interactions and is thus suitable to describe a very earlyUniverse dominated by such particles. The natural nextstep would be to include interactions of the inflaton withstandard model particles and with other particles, including

dark matter, into which inflatons should eventually decay.This inclusion would allow researchers to connect inflationwith the later Universe made up of both baryonic matter anddark matter. Next, researchers will need to expand the tem-poral range of the calculations to describe the emergence ofthe large-scale structure that we see in the Universe today.Although these later regimes would be very different fromthe primordial dark ages, some of the computational tech-niques employed in this work could prove useful to the task.

Finally, it’s important to determine whether there are anyobservable features that could allow researchers to probethis early postinflationary phase. Musoke et al. argue thatsuch features exist. They suggest that the growth of earlyfluctuations may produce gravitational fields sufficientlystrong to generate detectable gravitational waves. Thesewaves would appear as a stochastic background, much likethe cosmic microwave background. Depending on the en-ergy scale of inflation, these gravitational waves might fall ina frequency range accessible to existing detectors like LIGOand Virgo or to future space-based detectors like the LaserInterferometer Space Antenna (LISA), planned for launch in2034.

This research is published in Physical Review Letters.

REFERENCES[1] A. H. Guth, ‘‘Inflationary universe: A possible solution to the hori-

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solution of the horizon, flatness, homogeneity, isotropy and pri-mordial monopole problems,’’ Phys. Lett. B 108, 389 (1982).

[3] N. Musoke et al., ‘‘Lighting the dark: Evolution of the postinfla-tionary universe,’’ Phys. Rev. Lett. 124, 061301 (2020).

[4] L. Kofman et al., ‘‘Towards the theory of reheating after inflation,’’Phys. Rev. D 56, 3258 (1997).

[5] G. N. Felder and I. Tkachev, ‘‘LATTICEEASY: A program forlattice simulations of scalar fields in an expanding universe,’’Comput. Phys. Commun. 178, 929 (2008).

[6] A. V. Frolov, ‘‘DEFROST: A new code for simulating preheatingafter inflation,’’ J. Cosmol. Astropart. Phys. 2008, 009 (2008).

[7] M. A. Amin et al., ‘‘Oscillons after inflation,’’ Phys. Rev. Lett. 108,241302 (2012).

[8] K. D. Lozanov and M. A. Amin, ‘‘Equation of state and dura-tion to radiation domination after inflation,’’ Phys. Rev. Lett. 119,061301 (2017).

[9] M. A. Amin and P. Mocz, ‘‘Formation, gravitational clustering,and interactions of nonrelativistic solitons in an expanding uni-verse,’’ Phys. Rev. D 100, 063507 (2019).

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physics.aps.org c© 2020 American Physical Society 10 February 2020 Physics 13, 16