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\begin{document}

\title{densit\selectlanguage{ngerman}à universo\selectlanguage{english}}


\author[ ]{Marati Fortunato}

\affil[ ]{}
\vspace{-1em}


\date{}

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\begin{quote}
\textbf{Measuring the density of the Universe}

\textbf{by means of Newton's gravitational constant G}
\end{quote}

~ ~ ~ ~Marati Fortunato, ~Naples Italy

\par\null

\textbf{Article info:}

\textbf{Keywords:}

\textbf{Universe}

\textbf{density}

\par\null

\begin{quote}
Abstract:

Questions regarding the measure of Universe's density is of great
interest today. Several suggestions have been presented but the results
suffers from our lack of knowledge of the fraction of dark
matter~~present in and around galaxies.
\end{quote}

\subsection*{1. Introduction}\label{introduction}

\par\null

Methods to measure the actual density of the Universe are basically two:

\par\null

\textbf{The geometrical method} and \textbf{The computational method}

\par\null

\textbf{The geometrical method} is based on the converging/diverging
parallel lines. If the Universe is closed (density parameter \selectlanguage{greek}Ω\selectlanguage{english}0
\textgreater{} 1) the parallel lines converge and the observed density
of distant galaxies appears less than that expected by extrapolating the
local density of galaxies backwards in time. Oppositely if the Universe
is open (density parameter \selectlanguage{greek}Ω\selectlanguage{english}0 \textless{} 1) , the parallel lines
diverge and observed density of distant galaxies appears greater than
expected.

In the \textbf{computational method}~~ it is necessary to sample a
representative space region of the universe that is larger than the
scale on which the Universe becomes sufficiently homogeneous and measure
the masses of objects within the volume. The ratio of mass to volume
gives the density \selectlanguage{greek}\textbf{ρ} .

\selectlanguage{english}If the space region is the whole Universe, the volume is:

\textbf{V = ( 4 * pi *} \(R^3\)\textbf{) / 3}

where:

R is the radius of a sphere and it is equivalent at the distance of the
most far object.

This distance will be determined using the Hubble Law :

\textbf{}

\textbf{R = ~v}\textsubscript{\textbf{r}} \textbf{/
H}\textsubscript{\textbf{o}}

\textbf{}

where:

\textbf{~v}\textsubscript{\textbf{r}} is the recessional velocity,
typically expressed in km/s.

\textbf{H}\textsubscript{\textbf{o}}~is Hubble's constant or the ratio
of velocity to distance in the expansion of the Universe at the time of
observation.

\begin{quote}
The ''o'' on Ho~means the current value, since the Hubble ''constant''
changes with time (but it is the same everywhere in the Universe at a
given time).
\end{quote}

But since \textbf{vr = c * zmax :}

\textbf{}

\begin{quote}
\textbf{R = c * zmax / Ho}
\end{quote}

where:

\textbf{zmax~} is the maximum~redshift~at the R distance.

\textbf{c} is the light velocity.

Then:

\par\null

\begin{quote}
\emph{\textbf{V = (4 * pi / 3) * (c * z}\textsubscript{\textbf{max}}
\textbf{/
H}\textsubscript{\textbf{o}}\textbf{)}\textsuperscript{\textbf{3}}}\textbf{~}
\end{quote}

The mass of the objects in the selected space region is determined using
the virial theorem or one of its variants, which states that:

\textbf{}

\textbf{v}\textsuperscript{\textbf{2}} \textbf{= G * M / r}

\textbf{}

For hot X-ray gas in clusters of galaxies:

v ~is the typical thermal velocity and is determined by the temperature
of the gas.

r ~is an effective radius and~is determined using an angular
size~theta~and the distance D = c * z / Ho.

so:~ r = theta * D = theta * c * z / Ho.

M ~is the mass,.

G ~is Newton's gravitational constant.

Therefore the mass is given by:

M = r * v\textsuperscript{2} / G ~= ~theta * c * z *
v\textsuperscript{2} / ( G H\textsubscript{o})

Then:

\selectlanguage{greek}ρ \selectlanguage{english}= ~M / V = ( theta * c * z * v\textsuperscript{2} / (G *
H\textsubscript{o}) \textsubscript{} ) / (\emph{(4 * pi / 3) * (c *
z\textsubscript{max} / H\textsubscript{o})\textsuperscript{3}}~)

\par\null\par\null

2\textbf{. Calculation of the universe's density by means of}

\textbf{Newton's gravitational constant G and Universe age.}

The gravitational constant is a physical constant that is difficult to
measure with high accuracy.

George T. Gillies (1997),~''The Newtonian gravitational constant: recent
measurements and related studies'',~Reports on Progress in
Physics~\textbf{60}~(2): 151--225

The 2014~CODATA-recommended value of the gravitational constant is:

\par\null

\textbf{G} = 6,67408 * 10 \textsuperscript{-11}
\textbf{m\textsuperscript{3} / (sec\textsuperscript{2} * Kg )}

\par\null

\begin{quote}
''The 2014 CODATA Recommended Values of the Fundamental Physical
Constants'' ( \textbf{Last update:} June 25 2015).
Available:~http://physics.nist.gov/constants
\end{quote}

This constant is necessary to quantify ~gravitational~force between two
bodies and presumably it must ride on the global Universe's mass
allocated in the full Universe's volume at a definite moment.

Investigating its dimensions we notice:

\par\null

\textbf{G} = \textbf{m\textsuperscript{3} / (sec\textsuperscript{2} * Kg
)}.

\par\null

We make this hypothesis :

\par\null

\textbf{G = Universe's volume / (Universe's age \textsuperscript{2 ~}*
Universe's mass)}

\textbf{G = V / (t\textsuperscript{2} * M)}

\textbf{}

So we have:

\par\null

\textbf{M / V = 1 / (G * t\textsuperscript{2}) =} \selectlanguage{greek}ρ \selectlanguage{english}\textbf{= Universe's
density}

\par\null

In~physical cosmology, the~age of the universe~is the~time~elapsed since
the~Big Bang.

The~best measurement~of the age of the universe is~13.798\selectlanguage{ngerman}±0.037~billion
years

\textbf{(13.798±0.037×10}\textsuperscript{\textbf{9}}\textbf{~years
or~4.354±0.012×10}\textsuperscript{\textbf{17}}\textbf{~seconds)}within
the Lambda-CDM concordance model.

Planck Collaboration (2015). ``Planck 2015 results. XIII. Cosmological
parameters''

\par\null

The age of the Universe can be estimated by means of other methods too.

There are at least 3 ways:

\par\null

\begin{itemize}
\tightlist
\item
  The age of the chemical elements that gives a value of
  14.5+2.8/-2.2~Gyr.
\end{itemize}

(Nicolas Dauphas Nature~\textbf{435}, 1203-1205 (30 June 2005))

\par\null

\begin{itemize}
\tightlist
\item
  The age of the oldest star clusters that gives a value of 14.1 +/- 2.5
  Gyr
\end{itemize}

(Shinya Wanajo Astrophys.J. 577 (2002) 853-865)

\par\null

\begin{itemize}
\tightlist
\item
  The age of the oldest white dwarf stars that give a value of 12.8 +/-
  1.1 Gyr.
\end{itemize}

( Harvey B. Richer: Astrophys.J.574:L155-L158,2002)

\par\null

Replacing known values:

\par\null

\selectlanguage{greek}\textbf{ρ = 1 / (G * t}\selectlanguage{english}\textsuperscript{\textbf{2}}\textbf{) = 1 / (
(6.67408*10}\textsuperscript{\textbf{-11}}\textbf{) * ( 4.354 *
10}\textsuperscript{\textbf{17}}\textbf{)}\textsuperscript{\textbf{2}}
\textbf{)}

\textbf{}

\selectlanguage{greek}\textbf{ρ = 1 / ( (6.67408*10}\selectlanguage{english}\textsuperscript{\textbf{-11}}\textbf{) *
(18.957316*10}\textsuperscript{\textbf{34}}\textbf{) )}

\textbf{}

\selectlanguage{greek}\textbf{ρ = 1 / (126.52264356928 *
10}\selectlanguage{english}\textsuperscript{\textbf{23}}\textbf{) ~= ~0.0079 *
10}\textsuperscript{\textbf{-23 ~}}\textbf{Kg/m\textsuperscript{3}}
\textbf{= 7.9 * 10}\textsuperscript{\textbf{-26}}
\textbf{Kg/m\textsuperscript{3}}

\textbf{\textsuperscript{}}

\selectlanguage{greek}\textbf{ρ = 7.9 * 10}\selectlanguage{english}\textsuperscript{\textbf{-29}}
\textbf{g/cm\textsuperscript{3}}

\textbf{}

It is important to note that the achieved~\selectlanguage{greek}\textbf{ρ} \selectlanguage{english}is
the~total\textbf{~mass\_energy} density of the Universe. In other words,
it is the sum of a number of different components including both normal
(baryonic) and~\textbf{dark matter}~as well as the~\textbf{dark
energy}~.

\par\null\par\null

\textbf{3. Discussion and future work~}

\par\null

The achieved result gives a density value of about one order of
magnitude greater than the estimated values obtained by other methods.
So may be justified since we do not know accurately all the matter and
the energy existent in the whole universe.

It's notable that we do not have obtained a less value that would have
invalidated the assumptions we have made.

It is customary to express the density as a fraction of the density
required for the critical condition with the density parameter
\selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0}}.

The density parameter \selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0~}} is given by:

\par\null

\begin{quote}
\textbf{}

\selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0}} \selectlanguage{greek}\textbf{= ρ /
ρ}\selectlanguage{english}\textsubscript{\textbf{c}}

\textbf{}
\end{quote}

where (\selectlanguage{greek}\textbf{ρ}\selectlanguage{english}) is the~actual~density of the Universe and
(\selectlanguage{greek}\textbf{ρ}\selectlanguage{english}\textsubscript{\textbf{c}}) is the critical density (the
average density of matter required for the~Universe~to~just~halt its
expansion, but only after an infinite time).

This relation determines the overall geometry of the universe.

When the ratio is exactly~equal to 1, the geometry of the universe is
flat (Euclidean) and contains enough matter to halt the expansion but
not enough to recollapse it. If
~~\selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0~}}~is less than 1, the Universe is
open and will continue to expand forever. If
~\selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0}}~ is greater than 1, the Universe
is closed and will halt its expansion and recollapse.

The~current~critical density is approximately
\selectlanguage{ngerman}\textbf{(8.62±0.12)×10}\selectlanguage{english}\textsuperscript{\textbf{-27}}\textbf{~Kg/m}\textsuperscript{\textbf{3}}
\textbf{=
0.862x10}\selectlanguage{english}\textsuperscript{\textbf{-29}}\textbf{~g/cm}\textsuperscript{\textbf{3}}

Planck Collaboration (2015). ``Planck 2015 results. XIII. Cosmological
parameters''

\par\null

Using the value of \selectlanguage{greek}\textbf{ρ = 7.9 * 10}\selectlanguage{english}\textsuperscript{\textbf{-29}}~
to calculate the density parameter
~\selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0}}~ we have:

\par\null

\selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0}} \selectlanguage{greek}\textbf{= ρ /
ρ}\selectlanguage{english}\textsubscript{\textbf{c}} \textbf{}
\textsubscript{\textbf{~}}\textbf{= 7.9 *
10}\textsuperscript{\textbf{-29}} \selectlanguage{ngerman}\textbf{/ 0.862 ×
10}\textsuperscript{\textbf{-29}}\textbf{~ = 9.16}

\par\null

With the value of ~\selectlanguage{greek}\textbf{Ω}\selectlanguage{english}\textsubscript{\textbf{0}}~ = 9.16 the
Universe is closed.

\par\null

\textbf{References:}

\section*{how do astronomers measure the density of the
universe:}\label{how-do-astronomers-measure-the-density-of-the-universe}

\url{http://www.astro.ucla.edu/~wright/density.html}

\par\null

\textbf{Newton's gravitational constant G:}

''The 2014 CODATA Recommended Values of the Fundamental Physical
Constants'' ( \textbf{Last update:} June 25 2015).
Available\href{http://physics.nist.gov/constants}{p}:~\href{http://physics.nist.gov/constants}{http}\href{http://physics.nist.gov/constants}{://physics.nist.gov/constants}

\par\null

\subsection*{Universe's age:}\label{universes-age}

Planck Collaboration (2015). ``Planck 2015 results. XIII. Cosmological
parameters''

Bennett, C.L.;~et al.~(2013). ''Nine-Year Wilkinson Microwave Anisotropy
Probe (WMAP) Observations: Final Maps and
Results''.arXiv:1212.5225~{[}astro-ph.CO{]}.

\par\null

\subsection*{Critical density:}\label{critical-density}

Planck Collaboration (2015). ``Planck 2015 results. XIII. Cosmological
parameters''

\begin{quote}
''The 2014 CODATA Recommended Values of the Fundamental Physical
Constants'' ( \textbf{Last update:} June 25 2015).
Available:~\url{http://physics.nist.gov/constants}
\end{quote}

Zombeck's Handbook of Space Astronomy and Astrophysics.

Chapter 1, Page 11.

Chapter 10, Page 249.

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