 \leqslant \Gamma_{23} \leqslant 8$ (see
text for details).\label{fig:polytropic}}
\end{figure}

We investigate the structure of nonrotating neutron stars by solving
the Tolman-Oppen-heimer-Volkov equations. Since the central densities
of neutron stars can significantly exceed the regime of our neutron
matter calculations, we need to extend the 
EOS beyond this density. To this end, we employ a general polytopic
extension, where the pressure of neutron star matter is piecewise
given by $P(\rho) = \kappa\,\rho^{\Gamma}$, with mass density $\rho
=mn$~\citep{poly,Kai}. This is illustrated in Figure~\ref{fig:polytropic}.
At low densities we use the BPS crust EOS~\citep{BPS,Vautherin} up to
the transition density $\rho_{\rm crust}$. Figure~\ref{fig:polytropic}
shows that for densities $\rho_0/10$ to $\rho_0/2$ the BPS crust EOS
lies within the band predicted for the pressure of neutron star matter
based on the neutron matter results and incorporating beta equilibrium
following Section~\ref{sec:beta} [see also \citet{Kai}].
Therefore our results are insensitive to the particular choice of
$\rho_{\rm crust}$ within this region. In the following, we use
$\rho_{\rm crust} = \rho_0/2$, based on our results for the crust-core
boundary given by Figure~\ref{transsl}. The pressure from $\rho_{\rm
crust}$ to $\rho_1=1.1 \, \rho_0$ is given by the band predicted by
chiral EFT interactions. Our results are insensitive to the precise
value of $\rho_1$ in the vicinity of saturation density $\rho_0$, so
we have taken a conservative value for which the uncertainty band of
the microscopic neutron matter calculations is reasonable.

For the extension beyond $\rho_1$, we use three polytropes with
exponents $\Gamma_1, \Gamma_2$ and $\Gamma_3$, which make it possible
to vary the soft- or stiffness of the EOS in the density regions
1:~$\rho_1 \leqslant \rho \leqslant \rho_{12}$; 2: $\rho_{12}
\leqslant \rho \leqslant \rho_{23}$, and 3: $\rho \geqslant
\rho_{23}$, respectively (see Figure~\ref{fig:polytropic}). For
densities just above $\rho_1$, the EOS is still rather well
constrained by the neutron matter calculations. The band predicted for
the pressure of neutron star matter at $\rho_1$ corresponds to values 
of $\Gamma$ in the range $2.25 - 2.5$. Therefore, we take a restricted
range for the first polytropic exponent $1.0 \leqslant \Gamma_1
\leqslant 4.5$. We vary the value of all $\Gamma_i$ in steps of
$0.5$. At intermediate densities we allow for the possibility of a
phase transition and take a broad range $0 \leqslant \Gamma_2
\leqslant 8$. Finally, for densities beyond $\rho_{23}$ we allow for
$0.5 \leqslant \Gamma_3 \leqslant 8$.  We exclude the value 0 for this
density region in order to avoid artefacts connected with a phase
transition up to arbitrarily high density. For the densities between
polytropes, $\rho_{12}$ and $\rho_{23}$, we take $1.5 \, \rho_0
\leqslant \rho_{12} < \rho_{23} < \rho_{\rm max}$ with a step size of
$\rho_0/2$. We will show in the next section that the maximal density
for our suite of EOSs of neutron stars is $\rho_{\rm max} \approx 8.3
\, \rho_0$.

The general polytropic extension leads to a very large number of EOSs,
which cover all possible pressures in the grey region
in Figure~\ref{prho_extension}. Furthermore we emphasize that
this strategy is very general. It is based on a well defined
uncertainty band at nuclear densities and does not rely on assumptions
about the nature of the constituents of neutron star matter and their
interactions at higher densities. The values of $\Gamma_i$ and
$\rho_{ij}$ are limited by nuclear physics and observation. In the
following, we will demonstrate that the recent observation of a $1.97
\pm 0.04 \, M_{\odot}$ neutron star~\citep{Demorest} puts rather tight
constraints on the EOS at high densities and the radii of neutron
stars. We note that our results agree with those of our first study
\citep{Kai}, which used only two polytropes. This shows that the
general extension is robust and the conclusions would not change
significantly if additional polytropes were introduced to characterize
the pressure at high densities.

\section{Constraints on the nuclear equation of state and neutron stars}
\label{sec:constraints}

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.725,clip=]{prho_extensions_M197.pdf}
\hspace{2mm}
\includegraphics[scale=0.725,clip=]{prho_extensions_M24.pdf}
\end{center}
\caption{(Color online) Pressure $P$ of neutron star matter as a
function of the mass density~$\rho$. The blue band at lower densities
represents the pressure predicted by the neutron matter results
of Figure~\ref{p_nobeta} with beta equilibrium incorporated as described 
in Section~\ref{sec:beta}. The grey area shows the pressure accessed by
the possible polytropic extensions. The lighter 
blue band is the envelope of the general 
polytropic extensions that are causal and support a neutron
star of mass $\widehat{M} = 1.97 \, M_\odot$ (left panel) and $\widehat{M} =
2.4 \, M_\odot$ (right panel). We also show the individual EOSs within
these bands: The lines denote EOSs with central densities
$\rho_c \leqslant 2.5 \, \rho_0$ (yellow), for $2.5 \, \rho_0 < 
\rho_c \leqslant 5 \, \rho_0$ (orange), and for $\rho_c > 
5 \, \rho_0$ (red).\label{prho_extension}}
\end{figure}

The piecewise polytropic extension described in the previous section
is used to generate a very large number of equations of state that
cover the pressure-density plane at higher densities. We solve the
Tolman-Oppenheimer-Volkov equations for each of these EOSs and retain
only those that fulfill the following two constraints:
\begin{itemize}
\item[1.)] the speed of sound remains smaller than the speed of light
for all densities relevant in neutron stars: $v_s(\rho) =
\sqrt{dP/d{\mathcal E}} \, c \leqslant c$, where ${\mathcal E}$ is the 
energy density.
\item[2.)] the EOS supports a neutron star mass $M = \widehat{M}$, 
the mass of the heaviest neutron star observed.
\end{itemize}

We consider each EOS up to densities at which the maximal neutron star
mass is reached or the EOS becomes acausal, whichever density is
smaller. In Figure~\ref{prho_extension} we present the individual EOSs
that fulfill both constraints for two cases: $\widehat{M} = 1.97 \,
M_\odot$ (left panel), the mass of the heaviest known neutron
star~\citep{Demorest}, and $\widehat{M} = 2.4 \, M_\odot$ (right
panel), an estimated mass of the black widow pulsar
B1957+20~\citep{Kerkwijk}. However, since the uncertainties of the
latter determination are large, the $\widehat{M} = 2.4 \, M_\odot$
constraint should be considered as a hypothetical mass, which is used
here to probe the sensitivity of our results to the constraint from
observations.

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.8,clip=]{prho_compare.pdf}
\end{center}
\caption{(Color online) Constraints for the pressure $P$ of neutron 
star matter as a function of mass density $\rho$ compared to EOSs
commonly used to model neutron stars~\citep{LP}. The color coding of
the bands is as in Figure~\ref{prho_extension}.\label{prho_compare}}
\end{figure}

The blue band at lower densities in Figure~\ref{prho_extension}
represents the pressure predicted for matter in beta equilibrium as
described in Sections~\ref{sec:neutron_matter} and
~\ref{sec:beta}. The lighter blue bands give the allowed EOS range,
which is the envelope of the allowed polytropes at higher
densities. The color