**7.4. Anthropic interpretation of the cosmological constant**

The anthropic principle
[264,
265]
is an interpretational paradigm which argues that, while discussing the
origin of physical phenomena and the values of constants of nature, we
must recognize the fact that only certain combination and range
of values will lead to the existence of intelligence observers in the
universe who could ask questions related to these issues. This paradigm
has no predictive power in the sense that none of the values of the
cosmological parameters were ever predicted by this method.
^{(1)} In fact some
cosmologists have advocated the model with
_{NR} = 1,
_{} = 0
strongly and later - when observations indicated
_{}
1 -
have advocated the anthropic interpretation of cosmological constant with
equal fluency. This is defended by the argument that not all guiding
principles in science (Darwinian evolution, Plate tectonics, ....) need
to be predictive in order to be useful. In this view point, anthropic
principle is a back drop for discussing admittedly complicated
conceptual issues. Within this paradigm there have been many attempts
to explain (after the fact) the values of several fundamental constants
with varying degree of success.

In the context of cosmological constant, the anthropic interpretation works as follows. It is assumed that widely disparate values for the constants of nature can occur in an ensemble of universes (or possibly in different regions of the universe causally unconnected with each other). Some of these values for constants of nature - and in particular for the cosmological constant - will lead broadly to the kind of universe we seem to live in. This is usually characterized by formation of: (i) structures by gravitational instability, (ii) stars which act as gravitationally bound nuclear reactors that synthesize the elements and distribute them and (iii) reasonably complex molecular structures which could form the basis for some kind of life form. Showing that such a scenario can exist only for a particular range of values for the cosmological constant is considered an explanation for the value of cosmological constant by the advocates of anthropic principle. (More sophisticated versions of this principle exist; see, for example [266], and references cited therein.)

The simplest constraint on the cosmological constant is that it should
not be so high as to cause rapid expansion of
the universe early on preventing the formation of galaxies
[267].
If the energy density of the cosmological constant has to be less than
that of energy density of matter at the redshift
*z*_{gal}(
4) at
which galaxy formation takes place,
then we must have

(115) |

This gives a bound on
_{}
which is "only" a couple of orders of magnitude larger than what is
observed.

More formally, one could ask:
What is the most probable value of
_{} if
it is interpreted as the value that would have been observed by the largest
number of observers
[268,
269]?
Since a universe with
_{}
_{NR} will have
more galaxies than one with a universe with
_{}
10^{2}
_{NR},
one could argue that most observers will measure a value
_{}
_{NR}. The
actual probability *dP* for measuring a particular value for
_{}
in the range
(_{},
_{} +
*d*_{}) is the product
(*dP* / *d*_{}) = *Q*(_{})
(_{}) where
is the a priori
probability measure for a specific value of
_{} in
a member of
an ensemble of universes (or in a region of the universe) and
*Q*(_{}) is
the average
number of galaxies which form in a universe with a given value of
_{}.
There has been
several attempts to estimate these quantities (see, for example,
[270,
271]) but all of them
are necessarily speculative.
The first - and the most serious - difficulty with this approach
is the fact that we simply do not have any reliable way of estimating
; in fact, if we
really had a way of calculating it from a fundamental theory, such a
theory probably
would have provided a deeper insight into the cosmological constant
problem itself. The second issue has to do with the dependence of the
results on other parameters which describe the cosmological structure
formation (like for example, the spectrum of initial perturbations). To
estimate *Q* one needs to work in a multi parameter space and
marginalize
over other parameters - which would involve more assumptions regarding
the priors. And finally, anthropic paradigm itself
is suspect in any scientific discussion, for reasons mentioned earlier.

^{1} Some advocates of the anthropic
principle cite
Fred Hoyle predicting the existence of excited state of carbon nucleus,
thereby leading to efficient triple alpha reaction in stellar
nucleosynthesis, as an example of a prediction from anthropic principle;
it is very doubtful whether Hoyle applied anthropic considerations in
arriving at this conclusion.
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