New Constraints on the Variable Equation of State Parameter from X-Ray Gas Mass Fractions a

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NEW CONSTRAINTS ON THE VARIABLE EQUATION OF STATE PARAMETER FROM X-RAY GAS MASS FRACTIONS AND SNE Ia J.V.CUNHA ?,L.MARASSI ?and R.C.SANTOS ?February 5,2008Abstract Recent measurements are suggesting that we live in a ?at Universe and that its present accelerating stage is driven by a dark energy component whose equation of state may evolve in time.Assuming two di?erent parameterizations for the function ω(z ),we constrain their free parameters from a joint analysis involving measurements from X-Ray luminosity of galaxy clusters and SNe type Ia data.11Introduction In the framework of general relativity,the present accelerating stage of the Universe (as indicated by SNe type Ia observations)can be explained by assuming the existence of a substantial amount of an exotic dark energy component with negative pressure,also known as

quintessence[1,2].The existence of this extra component ?lling the Universe (beyond the cold dark matter)has also been indirectly suggested by independent studies based on ?uctuations of the 3K relic radiation,large scale structure,age estimates of globular clusters or old high redshift objects,as well as by the X-ray data from galaxy clusters[3,4].

A cosmological constant (Λ),the oldest and by far the most natural candidate for dark energy,faces some theoretical di?culties.The most puzzling of them is the so-called cosmological constant problem:the present cosmological upper bound,Λo /8πG ～10?47GeV 4,di?ers from natural theoretical expectations from quantum ?eld theory,～1071GeV 4,by more than 100orders of magnitude.Actually,such a problem has also inspired many scenarios driven

by aΛ(t)or a time varying decaying vacuum with constant equation of state[5].Among the remaining candidates to dark energy,the most promising ones lead to a time dependent equation of state(EOS),usually associated to a dynamical scalar?eld component.Such quintessence models may also parametrically be represented by an equation of state,ω(z),as proposed by Cooray and Huterer[6],as well as the one discussed by Linder[7],and,independently,by Padmanabhan and Choudhury[8].In principle,the time variation of the EOS parameter,ω(z)≡p/ρ,may allow a clear distinction between a cosmological constant model and the one driven by a rolling scalar?eld.

In actual fact,the exploration of the expansion history of the universe usingω(z)gave origin to a heated debate with growing interest in the recent literature.The SNe type Ia test is the most promising one related to this subject.However,Maor et al.[9],and Weller and Albrecht[10],have also observed that in order to constrain the evolution of the EOS with SNe observations,it is necessary to use a tight prior on the mean matter density of the Universe.

A natural way to circumvent such a problem is to consider the constraints on the density parameter from measurements of the X-Ray luminosity of galaxy clusters together in a joint analysis involving SNe Ia observations.

In this work we investigate the cosmological implications from X-ray of galaxy clusters and SNe data by considering two di?erent classes of EOS evolving with redshift.In the?rst scenario (hereafter Model1),the EOS parameter is de?ned by[6]

Model1:ω(z)=ωo+ω1z,(1) whereas in the second,the EOS parameter reads[7,8]

ω1z

Model2:ω(z)=ωo+

2Basic Equations

In what follows it will be assumed that the Universe is?at and its dynamics is driven by a pressureless cold dark matter(CDM)?uid plus a quintessence component.Both components are separately conserved and the EOS parameter of the quintessence component is represented by one of the parameterizations appearing in the introduction(see Eqs.(1)and(2)).By integrating the energy conservation laws for each component and combining the result with the FRW equation,it is straightforward to show that the Hubble parameter for both models can be written as:

H2Model1=H2o ?M(1+z)3+(1??M)(1+z)3(1+ω0?ω1)e3ω1z ,(3) and

H2Model2=H2o ?M(1+z)3+(1??M)(1+z)3(1+ω0+ω1)e?3ω1(z/1+z) ,(4) where the subscript“o”denotes a present day quantity and?M is the CDM density parameter.

On the other hand,the?rst attempts involving gas mass fraction as a cosmological test were originally performed by Pen[12]and Sasaki[13],and further fully developed by Allen et al.[4] who analyzed the X-ray observations for six relaxed lensing clusters observed with Chandra in the redshift interval0.1

f gas(z i)=b?b

D D

E A(z i)

1.5

,(5)

where b is a bias factor motivated by gas dynamical simulations that takes into account the fact that the baryon fraction in clusters seems to be lower than for the universe as a whole,?b stands for the baryonic mass density parameter,with the term(2h)3/2representing the change in the Hubble parameter between the default cosmology and quintessence scenarios while the ratio

D SCDM

A (z i)/D DE A(z i)accounts for deviations in the geometry of the universe from the Einstein-de

Sitter CDM model.

In order to derive the constraints from X-ray gas mass fraction in the next section we shall use the concept of angular diameter distance,D A(z).Such a quantity is readily derived in the present context(see,for instance,Refs.[14]and[15]):

D D

E A=H?1o

x2H(x)

.(6)

where x=R(t)

w 1

0w 10Figure 1:Marginalized constraints on plane ω0and ω1from joint analysis of the Chandra f gas (z )and SNe Ia data shown above for models 1(left panel)and 2(right panel).The solid lines mark the 1,2and 3σcon?dence limits.See text for more details.

4Conclusion and Perspectives

Nowadays,the signature of a dark energy or quintessence component is the most impressive observational fact of cosmology.As remarked elsewhere,we are living at a special moment where the emergence of new kind of “standard cosmology”seems to be inevitable.In the last few years,a growing attention has been paid for models with a time varying EOS parameter ω(z ).With basis on this sort of cosmological scenario,we have discussed here two simple possible parameterizations of the EOS obeyed by quintessence models as recently presented in the literature.Our results suggest that it is worthwhile to use the estimates of the gas mass fraction from galaxy clusters in joint analysis with the SNe Ia data since the derived constraints for ?M (and other quantities)do not require any prior to this parameter.More important,we have also obtained constraints for w o and w 1which have not been obtained before without a prior in ?M .

The parameterizations seems to be more e?cient to explain these data set,once they get a lower χ2;however they also have an additional parameter,and that is worthy of a study with more speci?c statistical criteria (Akaike or Bayesian information criteria,for example).A more detailed analysis of this kind will be investigated in the near future.

Acknowledgements

The authors are grateful to Prof.J.A.S.Lima for helpful discussions.This work was supported by CAPES (Brazilian Research Agency).

References

[1]T.Padmanabhan,Phys.Rep.380,235(2003);P.J.E.Peebles and B.Ratra,Rev.Mod.

Phys.75,559(2003);J.A.S.Lima,Braz.J.Phys.34,194(2004).

[2]A.G.Riess et al.,ApJ607,665(2004).

[3]P.de Bernardis et al.,Nature404,955(2000);L.Knox and L.Page,PRL851366(2000);

A.H.Ja?e et al.,PRL86,3475(2001);C.L.Bennett et al.,ApJ Suppl.148,1(2003);

R.G.Carlberg et al.,ApJ.462,32(1996);A.Dekel,D.Burstein and S.D.M.White, In Critical Dialogues in Cosmology,edited by N.Turok World Scienti?c,Singapore(1997);

P.J.E.Peebles,in Formation of Structure in the Universe,edited by A.Dekel and J.P.

Ostriker,Cambridge UP,Cambridge(1999);J.A.S.Lima and J.S.Alcaniz,MNRAS317, 893(2000);J.S.Alcaniz,J.A.S.Lima and J.V.Cunha,MNRAS340,L39(2003);G.

Sethi,A.Dev and D.Jain,Phys.Lett.B624,135(2005).

[4]S.W.Allen,R.W.Schmidt and A.C.Fabian,MNRAS334,L11(2002).See also S.Ettori,

P.Tozzi and P.Rosati,A&A398,879(2003).

[5]M.Ozer and M.O.Taha,Phys.Lett.B171,363(1986);Nucl.Phys.B287776(1987);

Carvalho et al.,Phys.Rev.D46,2404(1992);J.A.S.Lima and J.M.F.Maia,Phys.Rev D49,5597(1994);J.A.S.Lima and M.Trodden,Phys.Rev.D53,4280(1996);J.A.S.

Lima,Phys.Rev.D54,2571(1996);J.V.Cunha,J.A.S.Lima and N.Pires,A&A390, 809(2002);J.V.Cunha,J.A.S.Lima and J.S.Alcaniz Phys.Rev.D66,023520(2002);

J.V.Cunha and R.C.Santos,IJMP D13,1321(2004);C.S.Camara et al.,Phys.Rev.

D69,123504(2004);J.S.Alcaniz and J.A.S.Lima,Phys.Rev.D7*******(2005).

[6]A.R.Cooray and D.Huterer,ApJ513,L95(1999).

[7]E.V.Linder,Phys.Rev.Lett.90,091301(2003).

[8]T.Padmanabhan and T.R.Choudhury,Mon.Not.R.Astron.Soc.344,823(2003).

[9]I.Maor,R.Brustein and P.J.Steinhardt,Phys.Rev.Lett.86,6(2001).

[10]J.Weller and A.Albrecht,Phys.Rev.D65,103512(2002).

[11]J.K.Erickson et al.,Phys.Rev.Lett.88,121301(2002).

[12]U.Pen,New Astronomy2,309(1997).

[13]S.Sasaki,Pub.Astron.Soc.Japan48,L19(2002).

[14]J.A.S.Lima,J.V.Cunha and J.S.Alcaniz,Phys.Rev.D68,023510(2003).

[15]J.V.Cunha,J.S.Alcaniz and J.A.S.Lima,Phys.Rev.D69,083501(2004).

[16]W.Freedman et al.,ApJ553,47(2001).

[17]S.W.Allen et al.,Mon.Not.R.Astron.Soc.,353,457(2004).

[18]D.Kirkman et al.,ApJS149,1(2003).

[19]D.Jain and A.Dev,astro-ph/0509212(2005).