Cosmic explosions, life in the Universe and the Cosmological Constant

Galactic Gamma-Ray Bursts (GRBs) are copious sources of gamma-rays that can pose a threat to complex life. Using recent determinations of their rate and the probability of GRBs causing massive extinction, we explore what type of universes are most likely to harbour advanced forms of life. For this purpose we use cosmological N-body simulations to determine at what time and for what value of the cosmological constant ($\Lambda$) the chances of life being unaffected by cosmic explosions are maximised. We find that $\Lambda-$dominated universes favour the survival of life against GRBs. Within a $\Lambda$CDM cosmology, the parameters that govern the likelihood of life survival to GRBs are dictated by the value of $\Lambda$ and the age of the Universe. We find that we seem to live in a favorable point in this parameter phase space which minimises the exposure to cosmic explosions, yet maximises the number of main sequence (hydrogen-burning) stars around which advanced life forms can exist.

Why the value of the cosmological constant Λ is neither zero nor of the order of the Planck density (the Planck mass to the fourth power, M 4 Pl ) remains one of the deepest mysteries of nature. Refs [1,2] have argued that in order for observers (and thus cosmic structure) to exist, the value of Λ could not be larger than 10 −120 M 4 Pl . This was the first indication that the value of Λ could not be arbitrary and that requiring the existence of observers bounded Λ from above. Subsequent works (e.g., [3][4][5][6][7]) have firmed up this argument. However, to this date no argument has been given to provide a lower bound to Λ; in particular, it is not clear why Λ does not simply vanish 1 . Interestingly, the necessity to avoid massive life extinction events by GRBs can shed a new light on this issue.
GRBs are potentially catastrophic events for biological organisms 2 . In particular, copious flux of γ-ray photons with energies above 10 − 100 keV could destroy the ozone layer of a -habitable, Earth-like-planet, exposing living organisms to damaging UV radiation and compromising its habitability. This has led to the suggestion [9][10][11] that galactic GRBs have been responsible for some mass extinction events on Earth. Yet the rate or energy of nearby GRBs were not sufficient to avoid the emergence of observers. However, such GRBs take place more frequently at the inner parts of the Milky Way and may cause a serious problem for development of life there [12]. On Earth and in general, in the outskirts of large galaxies, the most luminous GRBs, the ones around the knee of 1 A lower bound to Λ based on the stability of atoms [8], applies only in the distant future for spatially flat universes. 2 In what follows when we refer to life we consider biological organisms which are sufficiently complex to act as observers. Our considerations will be for Earth-like planets where the UV protection provided by the atmosphere is due to an Ozone layer.
the luminosity function see Fig. 3 in [12], pose the greatest threat for the development of complex organisms 3 as they could cause catastrophic damage even if located in a sufficiently nearby satellite galaxy. The rate of GRBs within a given galaxy depends on the metallicity: most GRBs take place where metallicity < 0.3 solar and the stellar mass of the host galaxy is above 5 × 10 7 M [14,15]. Such low-metallicity environments are rare within the Milky Way and the ∼ 50kpc region around it. Small mass, low metallicity, Magellanic Cloud (SMC and LMC)-type galaxies are the typical host of GRBs and thus the most likely location for potentially damaging nearby GRBs. We explore next the rate of such catastrophic extra-galactic events.
The observed global GRB rate is 10 9 GRB/Gpc 3 /Gyr [16] 4 . GRB hosts have stellar masses between 5 × 10 7 and 10 10 M [14,15]. Integrating the stellar mass function of [17] in this mass range we find a stellar density of 10 16 M /Gpc 3 , yielding a rate of 10 −7 GRB/M /Gyr. This rate depends only on stellar physics and thus is independent of cosmology. Integrating (out to 200 kpc from the center) the dark matter profile from CDM simulations [18], the dark matter mass in satellites is 20% of the total halo mass (2×10 12 ) of the Milky Way. Since ∼ 1% of this mass is in stars [19] we obtain a stellar mass of 4 × 10 9 in the satellites 5 . Thus, we expect 400 GRB/Gyr in satellites. Using the observed GRB luminosity function [16] assuming an effective duration of 10s, we expect: 280 (70%) > 10 52 erg; 72 (18%) > 10 53.5 erg; 4(1%) > 10 54.5 and 0.5 − 1(0.15%) with energy of ∼ 10 55 erg. Ref [12] has shown that most likely there has been one GRB during the last Gyr with a fluence on Earth of 100kJ/m 2 ; this fluence is the value found by [21,22] for massive life extinction to take place. This event is believed to have caused the Orvidician extinction [23], which wiped out 85% of all species present on Earth at the time. Following [21,22] we take a fluence of 100kJ/m 2 to be the threshold: higher fluence would have catastrophic consequences for having observers 6 . Then the equivalent damaging distance from the center of the galaxy for a planet at 10 kpc from the center is 17kpc for 10 53.5 erg, 27 kpc for 10 54.5 and 50 kpc for 10 55 erg. This implies, conservatively, that a region of 20kpc (from the center of the host halo) should be devoid of GRB-hosting sub-halos for harbouring planets suitable to support observers. If there are no satellites in this region then there will be no damaging GRBs (of fluence 100kJ/m 2 ) with a rate much higher 7 than that on Earth. Since there is a nonzero probability of a ≥ 10 55 erg GRB/Gyr in the satellites which yields a lethal 100kJ/m 2 fluence at a distance of 50kpc, we also discuss this case.
The existence of many nearby satellites will have an additional effect. Many of these satellites in-fall into the main galaxy, bringing low metallicity material and triggering further star formation that will increase GRB production. We do not consider this enhanced rate in this work but it clearly makes development of intelligent life in cosmologies with numerous nearby satellites even more difficult than considered here.
Any inference of cosmological parameters ought to take into account the selection effects which have led us to observe the Universe from this particular vantage point [25]. Of particular interest is the value of Λ, which can govern the growth of regions around large galaxies devoid of LMC-type satellites. The accelerated expansion induced by a cosmological constant slows the growth of cosmic structures, and increases the mean inter-galaxy separation. This reduces the number of nearby satellites likely to host catastrophic GRBs. Below we quantify this effect.
Using N -body simulations, we search for halos with dark matter masses 10 11.5 < M/M < 10 12 that have no satellites of dark matter mass > 2 × 10 8 M within a radius of 20 (50) kpc (proper); we refer to these halos as isolated. In particular, we use the Millennium- 6 The amount of ozone depletion and DNA damage scales slowly with fluence: they are reduced by factors of 2 and 2.5 respectively by reducing fluence from 100 to 10 kJ/m 2 [24]. 7 About 8% of the satellites are within a radius of 20kpc for a Λ = 0 cosmology. Regardless of the cosmology, almost all satellites are within a radius of 50 − 100 kpc. To guide the eye, dashed lines show extrapolations to z = 0 due to limitations in re-mapping the Millenium-II simulation as this would have needed it to be run into the future. The number of lifeprotected regions is significantly larger in the Λ−dominated cosmologies. This is due to the effect of Λ at late times at clearing and smoothing virialised regions. Also remarkable is the fact that only below z ≈ 1 the number of isolated halos grows significantly. This is about 7 Gyr ago, not too dissimilar from the age of Earth.
II [26] publicly available dark-matter-only simulation 8 through the Millennium Database portal created by the Virgo Consortium [27]. We search for these isolated halos over all 37 available snapshots from z = 2 to z = 0 to determine the redshift evolution. We determine the number of subhalos N sh inside the search radius for each host halo (identified with a Friends-of-Friends ID number), which can be either empty (N sh = 0, isolated host halo) or not empty (N sh > 0, non-isolated host). The Millenium-II simulation is performed for a flat ΛCDM cosmology with parameters 9 : Ω m = 0.25, Ω b = 0.045, Ω Λ = 0.75, h = 0.73, σ 8 = 0.9, n s = 1. The size of the box is L = 100h −1 Mpc (comoving) on a side. The spatial resolution is 1h −1 kpc, so the search radius for subhalos is not affected by resolution effects. The mass resolution of the simulation is 6.89 × 10 6 h −1 M , so effectively our threshold on the mass of the satellite halo is verified by all subhalos resolved in the simulation, regardless of their mass. The host halo is required to have 30,000 to 100,000 particles, which corresponds to the halo mass bin stated above.
No public N-body simulations with the required mass resolution exist for other models than the "vanilla" ΛCDM. Fortunately, numerical simulations of one cosmology can be re-mapped into a different one accurately both for the dark matter field and the corresponding halos [28,29]. Using this algorithm we have re-mapped the results of the Millennium-II ΛCDM simulation to cosmologies with other values of Λ. Note that by keeping the geometry fixed (Ω m + Ω Λ = 1) and the early universe quantities (such as the physical matter density Ω m h 2 , the amplitude of primordial perturbations, and the baryon fraction) fixed the current value of Ω Λ specifies the cosmology.
It is challenging to keep track of changes of all other cosmological parameters and it is clear that other changes could mimic some of the effects that we emphasise here, for example significant amount of massive neutrinos would also suppress the growth of structure. However, it has been shown [30] that even modest modifications to a range of cosmological parameters leads to adverse consequences for the abundance of life. As such, we focus on changing only the value of the cosmological constant, while keeping the conditions of the early universe fixed. about 7 Gyr, not dissimilar from the age of Earth. As structure formation proceeds faster in the Λ = 0 universe, the amount of sub-structure grows faster and earlier: the number of isolated halos is much smaller than in the case of a Λ dominated universe.
In order for a particular galaxy to harbour life, it must reside within a habitable region of the parameter space. The halo mass should be in the range 10 11.5 < M/M < 10 12 as to give rise to a large galaxy like the Milky Way so that it has significant outer regions in which the GRB rate is low but there are sufficient number of stars. Additionally the galaxy should undergo sufficient chemical evolution so that its average metallicity is relatively large thereby reducing the GRB rate. Furthermore, the halo should be young enough that stars with mass M min have not left the main sequence. For M > 0.5M stars this corresponds to ( 50 Gyr), for M min > 0.7M this corresponds to 20 Gyr. For example for the model with Ω Λ = 0.5 the age of the Universe is ∼ 20 Gyr, by the time Ω Λ takes over the expansion, the universe is already too old and out of the habitable epoch for all main sequence stars above M min = 0.7M . For M min = 0.5M we estimate this to correspond instead to Ω Λ = 0.2 (ρ Λ /M 4 Pl = 1.2 × 10 −124 ). One can imagine waiting for a long time for the GRB rate to be sufficiently low. The current decrease of the GRB rate with time [16] at z < 1 is much flatter than the decrease in star formation rate (see their Fig. 9); extrapolating this rate implies that the star formation will be exhausted sooner than GRBs. By z ∼= −0.4-when flat universes with a cosmological constant that is 0.3-0.6 of that in the vanilla model become similar to the present Universe with Ω Λ = 0.75, the GRB rate will be down by a factor 2, while the star formation rate will be down by a factor 10. Since the decrease in star formation rate is the dominant effect, for simplicity in what follows we assume that the GRB rate remains constant. This approximation does not change qualitatively our argument.
In the range of ρ Λ that we have explored, and for M min = 0.7M , we find that the number density of habitable and isolated halos can be approximated by 11 The coefficients of the fit may change for different values of M min or different radii but the qualitative behaviour remains with a sharp decrease for Ω Λ < 0.2 (ρ Λ < 1.2 × 10 −124 ). For larger values of Λ this relation must flatten since virtually all MW-size halos are isolated for Ω Λ > 0.85. We also note that, always in the range of ρ Λ explored, the number of MW-sized halos in the same volume is roughly constant, implying that I(ρ Λ ) is roughly proportional to the fraction of isolated halos (for larger values of Λ when this fraction reaches unity it is assumed to become a constant). If we assume that, for the values of ρ Λ we considered so far, I(ρ Λ ) is proportional to the probability p(I|ρ Λ ), to infer the posterior probability for having a cosmological constant p(Λ|I) ∝ p(Λ)p(I|Λ) we must define a prior distribution p(Λ). Previous studies take a flat prior on ρ Λ based on the premise that its value may arise from the cancellation of much larger terms. However given the large uncertainty regarding the nature of dark energy, this might not be the case. A more conservative approach is to select an uninformative prior such that p(Λ) ∝ 1/ρ Λ (see also [6]). In this case one shall not consider negative values of the cosmological constant, although they are likely to generate even more inhospitable consequences. The other mechanism by which Λ influences the habitability of the Universe is the suppression of galaxy formation. This was studied in detail in [30] who used the fraction of baryons F (µ) residing in halos of mass µ ≡ ξ 2 M , where ξ is the matter density per photon, as a proxy for the habitability of the universe. They present the following prescription for the late time solution Here A and G ∞ can be considered fixed quantities, A = 5.59 and G ∞ = 1.43; we take s = 28 and ξ 4 Q 3 = 10 −124 11 For 50 kpc radius η = 0.1 and κ = 1.1 × 10 123 for ρ Λ in M 4 PL . For 20 kpc radius we find that a broken power law is a slightly better fit: κ 1 = 0.72 × 10 123 , η 1 = 0.52; κ 2 = 0.24 × 10 123 , η 2 = 0.72 respectively at low and high values of ρ Λ .
following [30] (see e.g., their Fig 7.). It is clear from Eq. 2 and e.g., Fig. 7 of [30] that this imposes a sharp suppression for ρ Λ 10 −122 M 4 Pl . In summary, where F (Λ) gives the number of suitable MW-size halos and describes the sharp upper cut-off imposed by eq. 2, and I(Λ) imposes a much slower suppression towards 0. In Fig. 2 we sketch the probability distribution function for ρ Λ , normalised to the Einstein-de-Sitter case, using the computed abundance of isolated Milky Way-size halos from Fig. 1. The low-end suppression is described by Eq. 1. For high values of ρ Λ the suppression discussed in [1,2,30], effectively due to the requirement of cosmic structure to form, becomes relevant. It is not surprising then that our universe has a value of Λ ≈ 1 (ρ Λ ∼ 10 −123 M 4 Pl ). It is instructive to compare to Fig.7 of [30] and notice that the allowed region for ρ Λ is now greatly reduced.
Since on average there is one isolated region in a patch of 10 Mpc radius (which, incidentally, is the mean inter galaxy cluster separation today), to ensure (at ∼ 3σ) that there is at least one habitable galaxy in the observable universe, the horizon size should be at least 40 Mpc i.e. not more than ∼ 100 times smaller than the current horizon size. Given our specifications for galactic habitability, both in terms of the required separations between galaxies, and the minimum age of the Universe which permits the formation of planets, a large universe is necessary for life to emerge. A loitering model with a finely tuned cosmological constant could satisfy the age requirement, but does not provide adequate inter-galactic spacing.
If our location in the Universe, and potentially the multiverse, is preferentially selected by the absence of nearby, GRB-hosting small halos, then we will expect to find that most Milky-Way sized galaxies beyond the Local Group possess a heightened number of LMC-like satellites. To some extent this has already been observed [31]. The reduced number of observed Milky-Way satellites compared to predictions derived from simulations is often attributed to the inefficiency of star formation within low mass halos. Yet this puzzle may also be partially resolved by our proposed selection effect.
Another interesting implication is that if the amplitude of fluctuations Q is increased, then halos of a given mass form earlier, but they do so in a very crowded environment. In previous investigations Q is one of the few cosmological parameters which could be enlarged by an order of magnitude without any clearly adverse effects (e.g., Fig.12 in [30]). Within the context of GRBs-limited habitability, there is likely to be much less freedom in this parameter.
In summary, we have shown that Λ plays a crucial role at creating habitable regions for galaxies in a habitable