7.2 Fine-tunings and determination of the anthropic range
As we have discussed in the previous sections, the outcome of many physical processes are strongly
dependent on the value of the fundamental constants. One can always ask the scientific question of what
would change in the world around us if the values of some constants were changed, hence doing some
counterfactual cosmology in order to determine the range within which the universe would
have developed complex physics and chemistry, what is usually thought to be a prerequisit
for the emergence of complexity and life (we emphasize the difficulty of this exercise when
it goes beyond small and local deviations from our observed universe and physics, see, e.g.,
[245] for a possibly life supporting universe without weak interaction). In doing so, one should
consider the fundamental parameters entering our physical theory but also the cosmological
parameters.
First there are several constraints that the fundamental parameters listed in Table 1 have to satisfy in
order for the universe to allow for complex physics and chemistry. Let us stress, in a non-limiting way, some
examples.
- It has been noted that the stability of the proton requires
. The anthropic
bounds on
,
and
(or on the Higgs vev) arising from the existence of nuclei,
the di-neutron and the di-proton cannot form a bound state, the deuterium is stable have
been investigated in many works [5, 6, 120, 145, 160, 161, 252
, 254], even allowing for nuclei
made of more than 2 baryon species [264]. Typically, the existence of nuclei imposes that
and
cannot vary by more that 60% from their observed value in our universe.
- If the difference of the neutron and proton masses where less that about 1 MeV, the
neutron would become stable and hydrogen would be unstable [442, 253] so that helium
would have been the most abundant at the end of BBN so that the whole history of the
formation and burning of stars would have been different. It can be deduced that [252]
one needs
so that the universe does not become all neutrons;
for the
reaction to be exothermic and
leading to a
finite domain.
- A coincidence emerges from the existence of stars with convective and radiative envelopes, since
it requires [80
] that
. It arises from the fact that the typical mass that separates
these two behavior is roughly
while the masses of star span a few decades around
. Both stars seem to be needed since only radiative stars can lead to supernovae,
required to disseminate heavy elements, while only convective stars may generate winds in their
early phase, which may be associated with formation of rocky planets. This relation while being
satisfied numerically in our universe cannot be explained from fundamental principles.
- Similarly, it seems that for neutrinos to eject the envelope of a star in a supernovae explosion,
one requires [80]
.
- As we discussed in Section 3.5, the production of carbon seems to imply that the relative
strength of the nuclear to electromagnetic interaction must be tuned typically at the 0.1% level.
Other coincidences involve also the physical properties, not only of the physical theories, but also of
our universe, i.e., the cosmological parameters summarized in Table 4. Let us remind some
examples.
- The total density parameter
must lie within an order of magnitude of unity. If it were
much larger the universe will have re-collapsed rapidly, on a time scale much shorter that the
main-sequence star lifetime. If it were to small, density fluctuations would have frozen before
galaxies could form. Typically one expects
. Indeed, most inflationary scenarios
lead to
so that this may not be anthropically determined but in that case inflation
should last sufficiently long so that this could lead to a fine tuning on the parameters of the
inflationary potential.
- The cosmological constant was probably the first one to be questioned in an anthropical
way [527
]. Weinberg noted that if
is too large, the universe will start accelerating
before structures had time to form. Assuming that it does not dominate the matter content
of the universe before the redshift
at which earliest are formed, one concludes that
. Weinberg [527] estimated
and concluded that “if it is
the anthropic principle that accounts for the smallness of the cosmological constant, then we
would expect the vacuum energy density
because there is no anthropic
reason for it to be smaller”. Indeed, the observations indicate
- Tegmark and Rees [486] have pointed out that the amplitude of the initial density perturbation,
enters into the calculation and determined the anthropic region in the plane
. This
demonstrates the importance of determining the parameters to include in the analysis.
- Different time scales of different origin seem to be comparable: the radiative cooling, galactic
halo virialization, time of cosmological constant dominance, the age of the universe today. These
coincidence were interpreted as an anthropic sign [65].
These are just a series of examples. For a multi-parameter study of the anthropic bound, we
refer, e.g., to [485] and to [243] for a general anthropic investigation of the standard model
parameters.