Astrophysical emissions on cosmic scales

1A homogeneous and isotropic universe?¶

The cosmological models that we have discussed postulate that the universe is homogeneous and isotropic on large scales, i.e. distances typically greater than 100 Mpc. Below this characteristic scale, the galaxies observed in wide-field optical surveys are distributed according to the cosmic web. The large structures that weave this web formed as a result of the growth of the fluctuations in the CMB. Due to the properties of gravitation, an initially ellipsoidal overdensity collapses preferentially along its smallest axis, resulting in a sheet (so called Zel’dovich pancake). These sheets can collapse in the form of filaments, which themselves converge towards galaxy clusters. The areas not populated by the cosmic web constitute the cosmic voids, which occupy most of space.

1.1Distance from local large-scale structures¶

The structures that form the cosmic web arrange themselves in a network resembling that of soap bubbles. A simplified crystallographic representation is given below, with $l$ the characteristic cell size and $w$ the typical diameter of clusters, or the cross-section of filaments. Cosmological simulations suggest a void occupancy rate of the order of $(1-w/l)³ = 75\%$ (see exercise below). Knowing that the characteristic radius of the clusters we observe is $w \approx 1\,$Mpc, we deduce a typical cell size of the order of $l \approx 10\,$Mpc.

The four types of structure (voids, sheets, filaments, clusters) are visible in the distribution of nearby galaxies. Our galaxy, its neighbour Andromeda at $0.75\,$Mpc and their satellites, such as the Large Magellanic Cloud at $50\,$kpc, form the Local Group. As illustrated in Figure 2, the Local Group is at the heart of the Local Sheet. This sheet is a planar structure with a diameter of ${\sim}\,10\,$Mpc, delimiting one side of the Local Void.

Opposite the galaxy NGC 253, about $16\,$Mpc from the Milky Way, is the Virgo cluster of galaxies, which has a rugby ball structure, as can be seen below. Galaxy clusters can contain several thousand galaxies.

Clusters of galaxies such as Virgo are connected to the cosmic web by filaments of galaxies, whose position in the celestial plane and radial velocity projection can be inferred by spectroscopy, as illustrated below.

On an even larger scale, typically $100\,$Mpc, the clusters are grouped into superclusters such as Laniakea, which hosts us.

1.2Mass of local large-scale structures¶

Not only can astronomers measure the position of galaxies in the sky, they can also measure the radial component of their velocity as well as their distance from the stars that calibrate the cosmic distance ladder. The velocity field of galaxies can thus be used to constrain the gravitational field of Laniakea or its companion supercluster Perseus-Pisces. The mass of the Laniakea supercluster is estimated at ${\sim}\,10^{17}\, M_\odot$ Tully et al., 2014 ^[1]. On a smaller scale, the mass of the Virgo cluster is estimated at ${\sim}\,10^{15}\, M_\odot$ Planck Collaboration et al., 2016. Such dynamical arguments also yield an estimate of the mass of the Local Group of galaxies at ${\sim}\,3 \times 10^{12}\, M_\odot$. The Local Group dynamical mass is dominated by that of the Milky Way, $(1. 2 \pm 0.2) \times 10^{12}\, M_\odot$ and Andromeda, $(1.8 \pm 0.5) \times 10^{12}\, M_\odot$ Benisty et al., 2022.

These masses are deduced using dynamical arguments such as the virial theorem (|kinetic energy| = |potential energy|/2 for a system at equilibrium) or using the projected trajectories of the tracers. Using stars as tracers, astronomers estimate the mass of the central black hole of the Milky Way, Sgr A*, at $(4.15 \pm 0.01) \times 10^{6}\, M_\odot$ GRAVITY Collaboration et al., 2019 (see Figure 6 ). The latter has a small contribution to the mass of our galaxy compared with the stars that make it up $(6 \pm 1) \times 10^{10}\, M_\odot$ Licquia & Newman, 2015. The mass of the Milky Way contained in the stars (and dust) is itself about 20 times less than its dynamic mass.

The mass deficit observed in the Milky Way is also inferred at the level of galaxy populations. The stellar mass of each galaxy can be estimated using near-infrared observations. At these wavelengths (around $1-3\,$µm), the emission from medium-sized stars like the Sun is high compared with that from dust, which emits mainly at wavelengths greater than $5\,$µm, and from more massive and younger stars, which emit mainly in the UV and blue band.

1.3Cosmic energy inventory¶

For an energy density today equal to the critical density $\rho_{c,0} = \frac{3 H_0^2 c^2}{8\pi G} = 1.36 \times 10^{11}\, h_{70}^2\, M_\odot\,\mathrm{Mpc}^{-3}$, i.e. $\varepsilon_{c,0} \approx 5.1\, h_{70}^2\,$GeV$\,$m^-3, where $h_{70} = H_0 / 70 \,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, we can see that only two thousandths of the universe’s energy budget is made up of stars. A detailed breakdown of the different energy budgets of the universe is shown below.

2Cosmic-scale engines behind astrophysical emissions¶

The processes responsible for most astrophysical emissions are star formation, the accretion of baryonic matter and the ejection of plasma jets by black holes, particularly supermassive black holes.

2.1Star formation¶

The evolution of baryonic matter and light emission follows the cosmic history of formation of stars. Their rate of formation per unit comoving volume is illustrated in the figure below.

In this figure, the redshift is converted to cosmological age using a flat ΛCDM cosmology, with Hubble constant $H_0 = 70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and dark energy density $\Omega_\Lambda = 0.7$, i.e. with $\Omega_\mathrm{m} = 0.3$. The look-back time is the counterpart of the cosmic age, i.e. $t_{L}(z) = \int_0^z \dd z' \left|\frac{\dd t}{\dd z'} \right|$, where

That is

with a negligible contribution from the energy density associated with the radiation, $\Omega_\mathrm{r}$.

The evolution of the star-formation rate, Ψ, determines the evolution of the fraction of the baryonic mass contained in stars and gas, as well as the enrichment of the interstellar medium in galaxies. For example, the mass density contained in stars, $\rho_\star$, can be calculated from the following conservation equation:

Here $R$ is the “return fraction”, i.e. the proportion of matter reinjected into the interstellar medium by stellar winds and explosions. The value of $R$ is estimated at $30-40\%$, depending on the initial distribution of stellar mass, called the initial mass function.

The evolution of star formation in the universe has led to an enrichment of interstellar media in atoms heavier than carbon, known as metals in thermal astrophysics. This enrichment is due to nucleosynthesis in the first stars from primordial hydrogen and helium (see pp chain and CNO cycle). These first stars, known as Population III stars, contributed to the reionisation of their environment during their short lifetime of a few million years. The most massive stars (${>}\,8\,M_\odot$) exploded in supernovae and ejected metals that fuelled subsequent generations of stars, and so on up to the Sun, which belongs to Population I. Gas, molecules - such as the most abundant H₂ and CO - and dust - in the form of carbonaceous and amorphous silicate grains - continued to accumulate in the interstellar medium throughout the first three billion years of the universe until the cosmic peak of star formation around a redshift $z \sim 2$, a period known as the “cosmic noon”. Since then, the universe has been populated mainly by long-lived Population I stars, typically $t_\odot \approx 10\,$Gyr for the Sun. Their average rate of formation has decreased as the reservoir of gas available in the interstellar medium has been depleted until today.

As with the mass of matter contained in stars today, the cumulative emission of nucleosynthesis processes is proportional to the integral of the cosmic star-formation rate.

2.2Accretion¶

Galaxies formed in a hierarchical fashion, first by the concentration of baryonic matter in their low-mass halos, then by merging with other galaxies in the cosmic web. The evolution of galaxies is accompanied by the growth of their central massive black hole. The mass of these black holes, $M_\bullet \approx 10^6 - 10^{10} M_\odot$, is about two thousandths of the stellar mass of the bulge of the host galaxy Ding et al., 2020. This relation illustrates the co-evolution of central black holes and their host galaxies. Many other correlations between observables support this link: for example the $M_\bullet - \sigma_\star$ relation between the black-hole mass and the dispersion of stellar velocities within spheroids (elliptical galaxy or central bulge for a spiral galaxy). The rate of accretion of matter by massive black holes thus follows an evolution, $\Psi_\mathrm{accr}$, comparable to that of the star-formation rate, with a density $2000-3000$ times lower.

Matter accretion around supermassive black holes, in particular around active galactic nuclei, is the second most important power-source for light emission after star formation. The energy released by accretion in the form of photons can be estimated using Soltan’s argument Soltan, 1982. The energy of a test particle accreted by a black hole from the last marginally stable orbit is only a fraction of the energy of its rest mass. The rest can be released in the form of radiation. The radiative efficiency of the accretion process, $\eta_\mathrm{accr}$, is defined as the ratio of the radiated power to the rate of mass-energy deposition in the disc, measured by an observer at infinity. Taking into account the deceleration of the black hole’s rotation by the accreted photons, Thorne calculates a radiative efficiency of $5.7\% < \eta_\mathrm{accr} < 30.8\%$ Thorne, 1974.^[2]

2.3Ejection¶

Around 10% of active galactic nuclei develop a jet on either side of the accretion disc Blandford et al., 2019. The presence of jets is deduced or observed in many astrophysical systems: collapse of a massive star leading to the formation of a long gamma-ray burst, binary of compact objects for a short gamma-ray burst, binary of a star and a white dwarf for a Nova, binary of a star and a compact object such as a black hole or a neutron star for a micro-quasar. Plasma ejection in the jets results from the conversion of electromagnetic energy at their base into kinetic energy at their termination. The kinetic energy of the jets can be estimated for some active galactic nuclei by the size of the radio cavities they form in the intracluster medium Fabian, 2012. Synthetic evolution models that attempt to reproduce these observations estimate that ${\sim}\,0.5\%$ of the mass energy associated with accretion is injected into the jets of active galactic nuclei Merloni & Heinz, 2008. However, the uncertainties surrounding the mechanisms of jet formation and the different accretion regimes mean that this ratio is merely a rough estimate of the blance between the kinetic energy and accreted energy.

The conversion of the Poynting flux (electromagnetic energy) into global plasma motion (kinetic energy) is accompanied by the acceleration of charged particles, for example by shock waves in the jets or at its boundary (shear), or by magnetic reconnection. Accelerated particles lose energy through radiative processes: synchrotron and inverse Compton for electrons and positrons, as well as production of pairs and pions for protons and nuclei. It is these radiative losses that produce the emission of jets from radio waves to gamma rays. The efficiency of converting kinetic energy into radiation is estimated at $10\%$ for jets of active galactic nuclei, gamma-ray bursts and microquasars.

Now that we have estimated the energy densities associated with the photon fields emitted by the three major astrophysical processes, we can compare them with observations.