Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the border lines between the regions of distinct solution type for
  solutions of the Friedmann equation satisfying the Hubble condition   It is well-known that there are four distinct basic types (two Big Bang
types, Lemaitre and Big Crunch type) for solutions of the general Friedmann
equation with positive cosmological constant, where radiation and matter do not
couple (see e.g. [2, p.7]. In that paper the system of case distinction
parameters contains a "critical radiation parameter" $\sigma_{cr}$. The present
note contains the constructive description of the so-called {\em border lines}
between Big Bang/Big Crunch type and Big Bang/Lemaitre type for so-called
Hubble solutions of the Friedmann equation by two smooth function branches,
expressing the cosmological constant as unique functions of the matter and
radiation density (which is considered as a parameter). These functions satisfy
simple asymptotic relations w.r.t. the matter density. They are constructed as
the solutions of the equation $\sigma=\sigma_{cr}.

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The categories of this paper are: hep-ex
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat