Recent Constraints on Models of Quintessence
M.Doran
We perform a comparison of the WMAP measurements with the predictions of quintessence cosmological models of dark energy. We consider a wide range of quintessence models, including: a constant equation-of-state; a simply-parametrized, time-evolving equation-of-state; a class of models of early quintessence; scalar fields with an inverse-power law potential. We also provide a joint fit to the CBI and ACBAR CMB data, and the type 1a supernovae.
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\newcommand{\omq}{\Omega_Q^{0}}
\newcommand{\omqls}{\Omega_Q^{ls}}
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\begin{document}
\vspace*{4cm}
\title{Recent Constraints on Models of Quintessence}

\author{M.Doran}

\address{Department of Physics \& Astronomy, Dartmouth College,
6127 Wilder Laboratory,
Hanover, NH 03755}

\maketitle\abstracts{
We perform a comparison of the WMAP measurements  
with the predictions of quintessence cosmological
models of dark energy. We consider a wide range of quintessence models,
including: a constant equation-of-state;  a simply-parametrized, time-evolving
equation-of-state;  a class of models of early quintessence; scalar fields with
an inverse-power law potential. We also provide a joint fit to the CBI and
ACBAR CMB data, and the type 1a supernovae.}


\begin{figure}[t]
%\scalebox{0.3}{\includegraphics{CMBdesc}}% Here is how to import EPS art
%\vspace{-0.4cm}
\begin{center}
\includegraphics[scale=0.45]{CMBdesc}
\caption{\label{fig:cmb_bw} 
The pattern of CMB anisotropy can reveal information about the quintessence
abundance ($\Omega_Q$), equation-of-state ($w$), and behavior of fluctuations
($\delta$). The three curves are examples of constant equation-of-state models
which differ little by eye, but are distinguished by the data. The red
($w=-0.5$) and blue ($-1.2$) curves are both low-$\chi^2$
CMB-indistinguishable, but distinct with respect to SNe. The black curve
($-0.8$), although it is consistent with the SNe data 
%and matches the  
%location and height of the first acoustic peak determined by WMAP,
%\cite{Page:2003fa}, 
is rejected by the CMB at the $3\sigma$-level.} 
\end{center}
\end{figure}


\section{Introduction}
We carry out an extensive analysis \cite{Caldwell:2003hz} of the CMB anisotropy and
mass fluctuation spectra for a wide range of quintessence\cite{Ratra:1987rm,Wetterich:fm,Caldwell:1997ii}
models. These models
are: (Q1) models with a constant equation-of-state, $w$, including $w<-1$; (Q2)
models with a simply-parametrized, time-evolving $w$; (Q3) early quintessence
models, with a non-negligible energy density during the recombination era; (Q4)
trackers described by a scalar field evolving under an inverse-power law
potential.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%% Parameter description
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The suite of parameters describing the cosmological models are split into
quintessence parameters, $\theta_{Q}$ and 
spacetime plus ``matter sector" variables, $\theta_{M}$, where $\theta_{M} =
\{\Omega_b h^2,$ $\Omega_{cdm} h^2,\, h,\, n_s,$ $A_S,\,\tau_{r}\}$. In order,
these are the baryon density, cold dark matter density, hubble parameter,
scalar perturbation spectral index, scalar perturbation amplitude, and optical
depth. We restrict our attention to spatially-flat, cold
dark matter models with a primordial spectrum of nearly scale-invariant density
perturbations generated by inflation.  \\[-3ex]
\section{Constant $w$ and phantom dark energy}
For a constant equation of state,
we have used the equivalence between a scalar field $\varphi$ with
potential $V(\varphi)$ and the equation-of-state $w$ in order to
self-consistently evaluate the quintessence fluctuations. For the range $w<-1$
we employ a {\it k}~essence model, keeping the sound speed (actually, this is
$d\omega^2/dk^2$) fixed at $c_s^2 = 1$. Since this model introduces only one
additional parameter beyond the basic set of spacetime plus matter sector
variables, we adopt a simplistic grid-based search for viable models. The
acceptance criteria for the Q1 models is based on a $\Delta\chi^2$-test. The
results of our survey of Q1 models are shown in
Figures~\ref{fig:HubbleFit},\,\ref{fig:BestFit}. Our basic conclusion from the overlapping
constraint regions is that there exist concordant models with $-1.25 \lesssim w
\lesssim -0.8$ and $0.25 \lesssim \Omega_{m} \lesssim 0.4$. 


 

\newcommand{\twowidth}{7.7cm}
\newcommand{\twoheight}{5cm}
\begin{figure}[t]
%\scalebox{0.4}{\includegraphics{HubbleFit}}% Here is how to import EPS art
\subfigure[The one-parameter family of best-fit models, which exploit the geometric
degeneracy of the CMB anisotropy pattern, is shown as the thick, red curve in
the $w-h$ plane. We have explored models in a six-dimensional cylinder in the
parameter space surrounding this ``best-fit line.''  The HST-Key Project
$1\sigma$ measurement of the Hubble constant is shown by the shaded band. ]{\label{fig:HubbleFit}\includegraphics[width=\twowidth,height=\twoheight]{HubbleFit}}
%\vspace{-0.3cm}
%\end{figure}
%\begin{figure}[h]
%\scalebox{0.4}{\includegraphics{BestFit}}% Here is how to import EPS art
\subfigure[The constraints on constant equation-of-state models due to CMB (WMAP, ACBAR,
CBI) and type 1a supernovae (Hi-Z, SCP) are shown. The starting point for our
parameter-search, the family of CMB-degenerate models, is shown by the thick,
black line.]{\label{fig:BestFit}\includegraphics[width=\twowidth,height=\twoheight]{BestFit}}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%% Q2-4 Figures
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\threescale}{0.22}
\begin{figure*}[t]
\subfigure[Q2: monotonic evolution $w(a)$]{\label{fig::monotonic1}
  \includegraphics[scale =\threescale,angle=-90]{f4.eps}
}
\subfigure[Q3: leaping kinetic quintessence]{ \label{fig::leaping1}
  \includegraphics[scale =\threescale,angle=-90]{f2.eps}
}
\subfigure[Q4: inverse-power law]{ \label{fig::ipl1}
  \includegraphics[scale =\threescale,angle=-90]{f6.eps}
}

\subfigure[Q2: monotonic evolution $w(a)$]{ \label{fig::monotonic2}
  \includegraphics[scale =\threescale,angle=-90]{f1.eps}
}
\subfigure[Q3: leaping kinetic quintessence]{ \label{fig::leaping2}
  \includegraphics[scale =\threescale,angle=-90]{f5.eps}
}
\subfigure[Q4: inverse-power law]{\label{fig::ipl2}
  \includegraphics[scale =\threescale,angle=-90]{f3.eps}
}
\caption{The results of our \mmc\ search of the multi-dimensional parameter
search, for models Q2-4, are illustrated in the figures above. In all cases, we
have marginalized over the suppressed parameters. The solid lines indicate the
$1,\,2,\,3\sigma$ contours based on comparison with the CMB (WMAP, ACBAR, CBI)
and type 1a supernovae (Hi-Z, SCP).} 
\end{figure*}

\section{Celestine quintessence:  $w(a) =w_0 + (1-a) w_1$}
%\vspace{0.2cm} \noindent{\it Q2:} 
We have examined quintessence models with an
equation-of-state that evolves monotonically with the scale factor, as $w(a) =
w_0 + (1-a) w_1$. This parametrization has been shown to be versatile in
describing the late-time quintessence evolution for a wide class of scalar
field models\cite{Linder:2002et}. Based on the degeneracy of models found for
Q1, we expect to find a two-dimensional family of equivalent best-fit models
with the same apparent angular size of the last scattering horizon, occupying a
plane in the $\{w_0,\,w_1,\,h\}$ space.  We find $w_0 < -0.75$ at the $2\sigma$
level, marginalizing over the suppressed five dimensional parameter space, as
illustrated in Figure~\ref{fig::monotonic1}. There, the shapes of the contours
indicate that current data can only distinguish between fast ($dw/da \gtrsim
0.5$) and slow evolution of $w(a)$, and offer only a weak bound on $w_1$.
However, in terms of $\omqls$ the relative quintessence density during recombination, 
we find $\omqls < 0.03$ at the $2\sigma$ level. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%% Q3 description
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Leaping kinetic term quintessence}
Leaping kinetic
quintessence features a 
non-canonical kinetic term that undergoes a sharp transition at late times,
leading to the current accelerated expansion\cite{Hebecker:2000zb}. At early
times the field closely tracks the cosmological background with $w=0$ during
matter domination, appearing as early quintessence\cite{Caldwell:2003vp}
before undergoing a steep transition towards a strongly negative
equation-of-state by the present day.  Since
$\omqls$ is not tied as closely to the expansion rate sampled by the
supernovae, compared to case Q2, the result is the weaker constraint $\omqls
\lesssim 0.1$, as shown in Figure~\ref{fig::leaping1}. Although the limit of a
cosmological constant can be approached in this model, the presence of early
quintessence will then require a sharp transition in the equation-of-state in
order to reach $w \to -1$. Consequently, we bar
models with $w \approx -1$ and non-negligible $\omqls$, as displayed in
Figure~\ref{fig::leaping1}. Next, because early quintessence
suppresses  the growth of fluctuations on small scales compared to large
scales, we find that comparable fluctuation spectra can be achieved by making a
trade-off between $n_s$ and $\omqls$ (see Figure~\ref{fig::leaping2}).\\[-4.5ex]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%% Q4 description
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Inverse power law quintessence}
%\vspace{0.2cm} \noindent{\it Q4:} 
Inverse-power law (IPL) models are the archetype quintessence
models with tracking property and acceleration\cite{Ratra:1987rm,Zlatev:1998tr,Steinhardt:nw}.  The potential is given by $V \propto
\varphi^{-\alpha}$, where the constant of proportionality is determined by
$\Omega_Q$. In certain supersymmetric QCD realizations of the IPL
\cite{Masiero:1999sq}, $\alpha$ is related to the numbers of color and flavors,
and can take on a continuous range of values $\alpha >0$. For $\alpha \to 0$,
however, inverse-power law models behave more and more like a cosmological
constant. From our analysis, we see that $\alpha \lesssim 1 - 2$, only a minor improvement
of earlier investigations\cite{Balbi:2000kj,Baccigalupi:2001aa,Doran:2001ty} using pre-WMAP data.

In   Figure~\ref{fig::ipl2} we plot the
likelihood contours in the $\Omega_m h^2 - \alpha$ plane: our results agree
with the best fit at $\Omega_m h^2 = 0.149$ for $\alpha=0$ or $w=-1$, but show
a tolerance for a wider range for $0 \le \alpha \le 2$. That is, the additional
degree of freedom in $\alpha$ means that the matter density for the IPL model
is not as well-determined from the peak position \cite{Page:2003fa} as compared
to the $\Lambda$ model. However, to maintain the peak at $\ell = 220$, we
observe that $\Omega_m h^2$ decreases slightly as $\alpha$ increases. \\[-3ex]


\section*{Acknowledgments}
This work was supported by NSF grant
PHY-0099543 at Dartmouth. We thank Pier Stefano Corasaniti for useful
conversations, and Dartmouth colleagues Barrett Rogers and
Brian Chaboyer for use of computing resources.
 
\begin{thebibliography}{99}


%\cite{Caldwell:2003hz}
\bibitem{Caldwell:2003hz}
R.~R.~Caldwell and M.~Doran,
%``Cosmic microwave background and supernova constraints on quintessence:  Concordance regions and target models,''
arXiv:astro-ph/0305334.
%%CITATION = ASTRO-PH 0305334;%%

 
\bibitem{Ratra:1987rm}
B.~Ratra and P.~J.~Peebles,
%``Cosmological Consequences Of A Rolling Homogeneous Scalar Field,''
Phys.\ Rev.\ D {\bf 37}, 3406 (1988).
%%CITATION = PHRVA,D37,3406;%%

 
\bibitem{Wetterich:fm}
C.~Wetterich,
%``Cosmology And The Fate Of Dilatation Symmetry,''
Nucl.\ Phys.\ B {\bf 302}, 668 (1988).
%%CITATION = NUPHA,B302,668;%%



\bibitem{Caldwell:1997ii}
R.~R.~Caldwell, R.~Dave and P.~J.~Steinhardt,
%``Cosmological Imprint of an Energy Component with General Equation-of-State,''
Phys.\ Rev.\ Lett.\  {\bf 80}, 1582 (1998).
%[arXiv:astro-ph/9708069].
%%CITATION = ASTRO-PH 9708069;%%


\bibitem{Linder:2002et}
E.~V.~Linder,
%``Exploring the expansion history of the universe,''
Phys.\ Rev.\ Lett.\  {\bf 90}, 091301 (2003).
%[arXiv:astro-ph/0208512].

\bibitem{Hebecker:2000zb}
A.~Hebecker and C.~Wetterich,
%``Natural quintessence?,''
Phys.\ Lett.\ B {\bf{497}}, 281 (2001). 
%[arXiv:hep-ph/0008205].

\bibitem{Caldwell:2003vp}
R.~R.~Caldwell, M.~Doran, C.~M.~Mueller, G.~Schaefer and C.~Wetterich,
%``Early quintessence in light of WMAP,''
[arXiv:astro-ph/0302505].

\bibitem{Zlatev:1998tr}
I.~Zlatev, L.~M.~Wang and P.~J.~Steinhardt,
%``Quintessence, Cosmic Coincidence, and the Cosmological Constant,''
Phys.\ Rev.\ Lett.\  {\bf 82}, 896 (1999).
%[arXiv:astro-ph/9807002].

\bibitem{Steinhardt:nw}
P.~J.~Steinhardt, L.~M.~Wang and I.~Zlatev,
%``Cosmological Tracking Solutions,''
Phys.\ Rev.\ D {\bf 59}, 123504 (1999).

\bibitem{Masiero:1999sq}
A.~Masiero, M.~Pietroni and F.~Rosati,
%``SUSY {QCD} and quintessence,''
Phys.\ Rev.\ D {\bf 61}, 023504 (2000).
%[arXiv:hep-ph/9905346].

\bibitem{Balbi:2000kj}
A.~Balbi, C.~Baccigalupi, S.~Matarrese, F.~Perrotta and N.~Vittorio,
%``Implications on quintessence models from MAXIMA-1 and BOOMERANG-98,''
Astrophys.\ J.\  {\bf 547}, L89 (2001).
%[arXiv:astro-ph/0009432].
%%CITATION = ASTRO-PH 0009432;%%

\bibitem{Baccigalupi:2001aa}
C.~Baccigalupi, A.~Balbi, S.~Matarrese, F.~Perrotta and N.~Vittorio,
%``Constraints on flat cosmologies with tracking quintessence from cosmic  microwave background observations,''
Phys.\ Rev.\ D {\bf 65}, 063520 (2002).
%[arXiv:astro-ph/0109097].
%%CITATION = ASTRO-PH 0109097;%%

\bibitem{Doran:2001ty}
M.~Doran, M.~Lilley and C.~Wetterich,
%``Constraining quintessence with the new CMB data,''
Phys.\ Lett.\ B {\bf 528}, 175 (2002).
%[arXiv:astro-ph/0105457].
%%CITATION = ASTRO-PH 0105457;%%


\bibitem{Page:2003fa}
L.~Page {\it et al.},
%``First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks,''
arXiv:astro-ph/0302220.
%%CITATION = ASTRO-PH 0302220;%%

\end{thebibliography}

\end{document}

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% End of moriond.tex  %
%%%%%%%%%%%%%%%%%%%%%%

astro-ph0310400.extracted_bib
\begin{thebibliography}{99}


%\cite{Caldwell:2003hz}
\bibitem{Caldwell:2003hz}
R.~R.~Caldwell and M.~Doran,
%``Cosmic microwave background and supernova constraints on quintessence:  Concordance regions and target models,''
arXiv:astro-ph/0305334.
%%CITATION = ASTRO-PH 0305334;%%

 
\bibitem{Ratra:1987rm}
B.~Ratra and P.~J.~Peebles,
%``Cosmological Consequences Of A Rolling Homogeneous Scalar Field,''
Phys.\ Rev.\ D {\bf 37}, 3406 (1988).
%%CITATION = PHRVA,D37,3406;%%

 
\bibitem{Wetterich:fm}
C.~Wetterich,
%``Cosmology And The Fate Of Dilatation Symmetry,''
Nucl.\ Phys.\ B {\bf 302}, 668 (1988).
%%CITATION = NUPHA,B302,668;%%



\bibitem{Caldwell:1997ii}
R.~R.~Caldwell, R.~Dave and P.~J.~Steinhardt,
%``Cosmological Imprint of an Energy Component with General Equation-of-State,''
Phys.\ Rev.\ Lett.\  {\bf 80}, 1582 (1998).
%[arXiv:astro-ph/9708069].
%%CITATION = ASTRO-PH 9708069;%%


\bibitem{Linder:2002et}
E.~V.~Linder,
%``Exploring the expansion history of the universe,''
Phys.\ Rev.\ Lett.\  {\bf 90}, 091301 (2003).
%[arXiv:astro-ph/0208512].

\bibitem{Hebecker:2000zb}
A.~Hebecker and C.~Wetterich,
%``Natural quintessence?,''
Phys.\ Lett.\ B {\bf{497}}, 281 (2001). 
%[arXiv:hep-ph/0008205].

\bibitem{Caldwell:2003vp}
R.~R.~Caldwell, M.~Doran, C.~M.~Mueller, G.~Schaefer and C.~Wetterich,
%``Early quintessence in light of WMAP,''
[arXiv:astro-ph/0302505].

\bibitem{Zlatev:1998tr}
I.~Zlatev, L.~M.~Wang and P.~J.~Steinhardt,
%``Quintessence, Cosmic Coincidence, and the Cosmological Constant,''
Phys.\ Rev.\ Lett.\  {\bf 82}, 896 (1999).
%[arXiv:astro-ph/9807002].

\bibitem{Steinhardt:nw}
P.~J.~Steinhardt, L.~M.~Wang and I.~Zlatev,
%``Cosmological Tracking Solutions,''
Phys.\ Rev.\ D {\bf 59}, 123504 (1999).

\bibitem{Masiero:1999sq}
A.~Masiero, M.~Pietroni and F.~Rosati,
%``SUSY {QCD} and quintessence,''
Phys.\ Rev.\ D {\bf 61}, 023504 (2000).
%[arXiv:hep-ph/9905346].

\bibitem{Balbi:2000kj}
A.~Balbi, C.~Baccigalupi, S.~Matarrese, F.~Perrotta and N.~Vittorio,
%``Implications on quintessence models from MAXIMA-1 and BOOMERANG-98,''
Astrophys.\ J.\  {\bf 547}, L89 (2001).
%[arXiv:astro-ph/0009432].
%%CITATION = ASTRO-PH 0009432;%%

\bibitem{Baccigalupi:2001aa}
C.~Baccigalupi, A.~Balbi, S.~Matarrese, F.~Perrotta and N.~Vittorio,
%``Constraints on flat cosmologies with tracking quintessence from cosmic  microwave background observations,''
Phys.\ Rev.\ D {\bf 65}, 063520 (2002).
%[arXiv:astro-ph/0109097].
%%CITATION = ASTRO-PH 0109097;%%

\bibitem{Doran:2001ty}
M.~Doran, M.~Lilley and C.~Wetterich,
%``Constraining quintessence with the new CMB data,''
Phys.\ Lett.\ B {\bf 528}, 175 (2002).
%[arXiv:astro-ph/0105457].
%%CITATION = ASTRO-PH 0105457;%%


\bibitem{Page:2003fa}
L.~Page {\it et al.},
%``First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks,''
arXiv:astro-ph/0302220.
%%CITATION = ASTRO-PH 0302220;%%

\end{thebibliography}