Sergei Bashinsky

Relic neutrinos — the neutrinos created in the Big Bang and now forming the “cosmic neutrino background” — are the second most numerous particles in the universe. Only the photons of the cosmic microwave background (CMB) somewhat outnumber them. The relic neutrinos have streamed unimpeded since the first second of the Big Bang, for hundreds of thousands years before the last scattering of the CMB photons. Interacting weakly but being neutral to electromagnetic and strong interactions, the relic neutrinos so far escape direct detection by particle detectors. However, the relic neutrinos also interact gravitationally. Hence they can be probed en masse by cosmological observations.

Of utmost importance for revealing relic neutrinos and determining their properties is identification of their non-degenerate observable cosmological signatures. These are the features in measurable characteristics of the universe that are produced by the neutrinos and that cannot be imitated by changing other cosmological parameters: the abundance of dark matter and dark energy, the rate of cosmological expansion, etc. Fortunately, relic neutrinos do provide distinctive non-degenerate signatures.

Specifically, I determined^[1] that of all the known particles and matter species of the Standard cosmological model only relic neutrinos shift additively^[1] the pronounced acoustic peaks in the CMB power spectrum, by shifting the temporal phase of the acoustic oscillations.

(“Acoustic oscillations" are the acoustic waves in the slightly inhomogeneous “photon-baryon" plasma of the early universe: the plasma of photons, electrons, and ions — all interacting before the electrons and ions “recombined" into a gas of neutral atoms, transparent to the photons, which then became the CMB. These acoustic waves had been sourced by the primordial inhomogeneities, expected from cosmological inflation. They produced the most prominent feature in the observed angular power spectrum of the CMB: its acoustic peaks.)

Paper [1] proved that (for the standard, inflation-generated cosmological inhomogeneities, called “adiabatic") the temporal phase of the acoustic oscillations is shifted by and only by the gravitational impact of the species of particles or other cosmic matter with the following two properties:
  (i) the species interact with the photon-baryon plasma only gravitationally;
 (ii) perturbations in the species propagate faster than the “acoustic speed", which is the speed of the acoustic waves in the photon-baryon plasma.
Of the known elementary particles, only neutrinos, freely streaming at nearly the speed of light, fulfill both conditions.

CMB observations have only recently reached the resolution required to study relic neutrinos:
• In 2015 the predicted^[1] neutrino-induced additive shift of the acoustic peaks in the CMB was observed by ESA’s Planck satellite [3,4]. The measurements of the magnitude of the shift in the Planck data confirmed [3,4] my theoretical prediction [1].
• In 2018 the same neutrino-induced shift was also detected^[5] in the baryon acoustic oscillations, which are the imprints of the acoustic oscillations in the photon-baryon plasma on the distribution of galaxies.
• The accuracy of determining neutrino properties with cosmological observations can still be improved dramatically by raising the angular resolution of maps of CMB temperature and polarization, the latter being particularly important^[1] for studying neutrinos.

Another detectable signature of relic neutrinos, also first shown in [1], is increased diffusion damping (a.k.a. Silk damping) of the CMB inhomogeneities on small scales. The increase of damping is caused by faster expansion of the radiation-dominated universe whose energy density is enhanced by the relativistic neutrinos.

(The mechanism for the increase of diffusion damping in the CMB by decoupled relativistic species, including light neutrinos, thus differs from more familiar smoothing of inhomogeneities in warm dark matter, e.g., in massive neutrinos, homogenized by particles streaming out of overdense regions into underdense ones.)

Only free-streaming relativistic particles shift the phase of the acoustic peaks. Complementary, the total radiation energy density affects diffusion damping, also increased by the relativistic species that contribute to the radiation density but do not stream freely. Thus, jointly, the phase shift and diffusion damping of the CMB power spectra let us measure the abundance of relativistic species and constrain their interactions.

The shift of the acoustic peaks and the increase of diffusion damping of CMB anisotropy constrain the density and interactions of any “dark radiation”: the standard neutrinos as well as other possible decoupled relativistic particles, e.g., light sterile neutrinos. Moreover, these and other found cosmological signatures^[2] of the dark radiation may expose^[2] a scalar field, called “quintessence”, as its energy is expected to track the radiation energy in the radiation-dominated universe, before the field manifests itself as the dark energy when the energy density of the matter in the expanding universe exceeds the redshifting energy density of the radiation.

Various types of invisible matter provide a large or dominant fraction of energy of the universe throughout its entire evolution: past, present, and future. These “dark” species—dark matter, dark energy, and dark radiation (the known neutrinos and possibly yet undiscovered relativistic species)—do interact with the regular matter and light gravitationally. This lets us unveil the dark species and probe their properties with cosmological observations of visible matter and light, including the CMB.

We can obtain particularly specific and detailed information about the dark species by examining evolution of their inhomogeneities, or “perturbations”. Inhomogeneous cosmological dynamics at various spatial scales depends on (so, tells us about) many more properties of the dark species than homogeneous—averaged over space, or “background”—dynamics does.

Many dark species, including neutrinos and dark energy, resist gravitational collapse. Inhomogeneities in such dark species can gravitationally impact visible matter and light only during their “horizon entry”; soon after the entry, the inhomogeneities become unobservable. Moreover, the cosmic microwave background (CMB)—the sharpest of today's cosmological probes—is most prominently affected by perturbations in any dark species (including massive and pressureless dark matter^[0]) during the horizon entry as well.

[Horizon entry of an inhomogeneity (of a perturbation) is the time when the expanding Hubble horizon had become comparable to the spatial size of the inhomogeneity. From the start of the Big Bang by the time of horizon entry light and other interactions had just traversed the inhomogeneity length.]

The standard computer codes, e.g., CAMB or its predecessor CMBFAST, let us compute the values of many observables for a given model of dark dynamics. Yet correct computation of observables does not automatically translate into correct understanding of which property of the dark species affects which observable. All or most of imprinting of inhomogeneities in the dark species on observables occurs during the horizon entry. Then evolution is dominated by general relativistic effects and looks very different, even qualitatively, when described in terms of different variables that quantify the inhomogeneities.

The so-called “gauge-invariant”^[1,2] or covariant^[3,4] approaches neither eliminate nor even alleviate this ambiguity. They only replace the choice of gauge—i.e., of coordinates for the curved perturbed spacetime—by the choice of gauge-invariant variables for quantifying inhomogeneities.

Most of the technically acceptable and popular descriptions do not reflect explicitly the cause-effect relationship between properties of dark species and features in measurable cosmological distributions that are affected by those properties. We can correctly compute observables with any description. Yet most descriptions, including those being taught in most cosmology classes and used most widely, lead to erroneous conclusions about the origin of crucial visible features.^[0]

In contrast, [5] proves mathematically that the descriptions of weakly inhomogeneous cosmological evolution that satisfy two simple and natural criteria render the objective causal dependencies directly. At the same time, the dynamical equations and their solutions in a formalism^[8] that fulfills these two criteria simplify remarkably.

Deeper than being an illustration, this is equivalent to the distinction between the heliocentric and geocentric descriptions of the solar system. In general relativity, no coordinate frame for arbitrarily curved spacetime is fundamentally preferred. Yet the inertial frames become physically preferred in the limit of weak gravity. Only in them do velocities of free objects remain unchanged without influence of other bodies. Hence any change of the velocities of planets, comets, etc. in an inertial frame is objectively caused by influence of other physical bodies: the Sun, other planets, etc.

One could specify the dynamics of the solar system and crunch it with computers in the non-inertial geocentric frame, adding the centrifugal and Coriolis forces. Yet the resulting description—the Ptolemaic system with epicycles—obscures the causal connection between the path of planets, comets, etc. that we see on the sky and the gravitating bodies that shape the observed path. Also, the non-inertial geocentric frame yields more complicated equations and unwieldy solutions.

Likewise, certain variables for cosmological perturbations become physically preferred in weakly inhomogeneous expanding space (weakly perturbed FLRW metric). These variables do not change without an objective—locally measurable and gauge-independent—source that causes the change.^[0,7] This criterion distinguishes the suggested, physically preferred variables from the traditional variables, used historically as seemingly “natural”, as the geocentric Ptolemaic view of the solar system once also seemed to be. As with the Copernican view, the requirement of “a change only by a cause" streamlines the formalism, simplifies the equations and their solutions substantially [e.g., slides 11-13 here]. Then the solutions—as proven in [0, 7]—manifestly point at the characteristics of the CMB and galaxy distributions that map to specific internal properties of the dark species.^[5] These solutions also reveal new signatures^[8] of the dark dynamics that are subtle yet non-degenerate, hence are particularly valuable.

The suggested variables^[8] are the physical perturbations of density, intensity, phase-space distribution, etc. on subhorizon scales—for microscopic, terrestrial, and galactic distances. Yet on the cosmological scales comparable to and exceeding the Hubble horizon, where general relativity becomes essential, these variables evolve substantially simpler than the traditional variables do: qualitatively simpler and by simpler equations [see slides 11-13]. They are well defined for any species—particles, fields, and any other—through the species’ energy and momentum. (They are also simply related to the “conserved” Bardeen curvature^[9] ζ, standard for quantifying superhorizon perturbations, and to its generalization^[10] for individual species.)

Then we find that some past views should be changed substantially for understanding the actual origin of even the most prominent and physically important features in the observed cosmological distributions.

For example, the traditional formalism “shows” that the pronounced enhancement of the first acoustic peak in the CMB angular power spectrum C_l relative to the Sachs-Wolfe plateau at small multipole values l is mainly caused by the “resonant radiation driving”^{[e.g., 11,12,13,14]}. This is incorrect. No resonant boost of the small-scale CMB power by radiation objectively exists^[0]. Indeed, if “resonant radiation driving” were real then the measured CMB anisotropy on small scales—at the first and higher acoustic peaks—would be sensitive to dynamics of the radiation. It is not the case: (click to expand)

We would not offer the Ptolemaic system as the primary description of planetary dynamics in today’s undergraduate astrophysics courses. In graduate cosmology too, it is time to stop explaining the general relativistic evolution that shapes the anisotropy of the CMB and large scale structure of the universe — the pillars of contemporary precision cosmology — by similarly misleading and technically cumbersome formalisms. Encouragingly, some cosmology groups at prominent universities are adopting^{[15, 16]} the more direct and simple approach.

There is overwhelming evidence from an array of independent cosmological observations that the expansion of our universe is accelerating. Empirical support for acceleration of the cosmological expansion is provided by observations of type Ia supernovae, CMB anisotropy, growth of the large scale structure of the universe, ages of galaxies versus their redshifts, etc. The cause of the acceleration, however, has so far no universally accepted theoretical explanation. The most popular alternatives are:
(i)  “dark energy”—a new form of matter with large negative pressure;
(ii)  “modified gravity”—deviation of gravitation laws from general relativity on cosmological scales.

Surprisingly, the task of empirically distinguishing these, superficially dissimilar, scenarios is extremely subtle. We cannot differentiate between them, even in principle, by measuring global characteristics of the universe and their change with redshift: the rate of cosmological expansion at various redshifts or the global curvature. Neither can we distinguish between dark energy and modified gravity by observing inhomogeneities in the cosmic microwave background (CMB) and in other visible matter on large scales where inhomogeneous evolution is linear.^[1]

Alikeness of detectable signatures of dark energy and modified gravity is deeply rooted. Suppose that the acceleration of the cosmological expansion is caused by modified gravity. Let the dynamics of the visible matter and light remain generally covariant, i.e., obey the standard equations of motion in somehow curved spacetime. Yet let the curvature of spacetime, and so the gravitational forces that act on the visible species, be not determined by the Einstein equations of general relativity.

Nevertheless, then we could still “explain” all cosmological observations with the Einstein equations by adding to the visible species some “effective” dark species.^[2]

Namely, any curved spacetime admits an effective energy-momentum tensor^[2] that relates to its curvature exactly as the Einstein equations prescribe. This tensor is covariantly conserved,^[2] as the energy-momentum tensor would be in general relativity. Thus even in a modified-gravity universe, not governed by general relativity, assumption of general relativity and dark species yields correct motion of the visible matter and light as long as the distribution of energy and momentum of the visible and hypothesized dark species is set by the effective energy-momentum tensor.

In particular, all the observable manifestations of dark energy with any possible internal dynamics and of modified gravity can be described by the same independent phenomenological parameters for the general conserved energy-momentum tensor.^[2] With this surprising finding, we may ask: Is general relativity even falsifiable?

It is. Definitive methods to distinguish modified gravity from dark energy via cosmological observations were originally explained in my paper [2]. Modified gravity can be detected by establishing that the dynamics of the effective “dark species”, introduced to reconcile the observations with the Einstein equations, should strongly depend on the distribution of visible matter: more strongly than possible from coupling of the dark and visible species only gravitationally.^[2] Modified gravity can also be detected by concluding that to explain observations with Einstein’s general relativity, the inferred “dark species” have to move superluminally.^[2] At last, general relativity can be falsified by nonstandard phenomenology of gravitational waves.

Anisotropy of the cosmic microwave background (CMB) contains rich information about the history and composition of the universe, from its birth in inflation to its presently accelerating expansion and ongoing formation of galaxy clusters. Some of the information in the CMB reflects the abundance and internal properties of dark species. The CMB decoupled around a redshift z≈1,100. However, the CMB sensitively probes the dark dynamics well after and—even more so—well before its decoupling, up to a redshift of the order of z∼10⁵.^[1] Indeed, inhomogeneities in any dark species gravitationally affected the CMB most of all when the inhomogeneities were entering the Hubble horizon, not when the CMB was decoupling.

For decoding these imprints, understanding of causal dependencies in the general relativistic interaction on large scales is crucial.

The cosmic-variance limit on the accuracy of determining physical quantities with CMB observations improves dramatically toward high redshift.^[1] There, the accuracy is limited only by the present angular resolution of CMB detectors and noise-removal techniques.

Cosmological evolution in recent time—i.e., at small redshift—is highly nonlinear. Semi-phenomenological approaches, such as proposed by Press and Schechter and extended by others (notably, Bond, Cole, Efstathiou, and Kaiser), let us understand nonlinear evolution intuitively. Its accurate study however requires numerical simulations. It is worthwhile to explore:
(a)  how various limitations and approximations of the simulations affect their accuracy and reliability;
(b)  how to understand results of the simulations in terms of simple dynamics-based considerations;
(c)  how to develop the most accurate and fast simulation for given computational resources.

Cosmological inhomogeneities—or perturbations—are typically analyzed through their Fourier decomposition into harmonic waves. An alternative and useful approach is to decompose the inhomogeneities into spatially localized Green's function. These methods are complementary, each offering its own advantages and highlighting different physical phenomena. The Green's function technique is more explicit for phenomena that involve spatial transport of particles or energy. For example, Green's functions led to new results in studies of CMB acoustic waves, freely streaming neutrinos, and relic gravitational waves.

In particular, Green's functions yielded the first analytic solution for evolution of cosmological perturbations that accounts for decoupled relativistic neutrinos (for both scalar^[1] and tensor^[2] modes). They also helped notice that the acoustic waves in the CMB produce a narrow feature^[3] in the real-space auto-correlation functions of the CMB temperature and galaxy density.^[3]

I also applied my technique of plane-parallel cosmological Green's functions to determine the evolution of relic gravitational waves that interact with relic neutrinos. This produced the first analytic solution^[1] for their coupled evolution. The analytic solution confirmed noticeable (about 40%) suppression, found originally by Steven Weinberg numerically^[2], of the tensor—primordial gravitational waves—power spectrum on short scales by the streaming neutrinos.

The Green's function solution first determined that neutrinos do not affect the phase of the oscillations in the tensor power spectrum—a characteristic of primordial gravitational waves that may be measured with CMB polarization. The solution revealed yet another previously unknown result^[1]: the phase of the oscillations in the tensor power spectrum could deviate from zero only due to a substance that supports physical superluminal propagation.