Apparent horizons in Clifton-Mota-Barrow inhomogeneous universe
Vincenzo Vitagliano, Valerio Faraoni, Thomas P. Sotiriou, Stefano Liberati
aa r X i v : . [ g r- q c ] F e b June 8, 2018 8:49 WSPC - Proceedings Trim Size: 9.75in x 6.5in BH˙Vitagliano˙DEF APPARENT HORIZONS IN CLIFTON-MOTA-BARROWINHOMOGENEOUS UNIVERSE
VINCENZO VITAGLIANO , VALERIO FARAONI , THOMAS P. SOTIRIOUand STEFANO LIBERATI CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico, Universidade T´ecnica deLisboa - UTL, Av. Rovisco Pais 1, 1049 Lisboa, Portugal Physics Department and
STAR
Research Cluster, Bishop’s University,Sherbrooke, Qu´ebec, Canada J1M 1Z7 SISSA, Via Bonomea 265, 34136 Trieste, Italy andINFN, Sezione di Trieste, Italy
We analyze the apparent horizon dynamics in the inhomogeneous Clifton-Mota-Barrow solution of Brans-Dicke theory. This solution models a central matter configura-tion embedded in a cosmological background. In certain regions of the parameter spacewe find solutions exhibiting dynamical creation or merging of two horizons.
1. Introduction
Scalar-tensor theories lead to a spacetime-dependent effective gravitational coupling G eff . In theories with this characteristic, the problem of understanding the behaviourof a local system embedded in a cosmological environment (cf. the McVittie solu-tion of general relativity which is not yet well understood), is of particular interest.This motivated the study of inhomogeneous analytical solutions of scalar-tensor the-ories of gravity.The Clifton–Mota–Barrow spherically symmetric and time-dependent metric is ds = − A ( ̺ ) α dt + a ( t ) "(cid:18) m α̺ (cid:19) A ( ̺ ) α ( α − α +2) ( d̺ + ̺ d Ω ) , with A ( ̺ ) = 1 − m α̺ m α̺ , α = s ω + 2)2 ω + 3 , a ( t ) = a (cid:18) tt (cid:19) ω − γ )+23 ω γ (2 − γ )+4 ≡ a ∗ t β , where γ − S BD = Z d x √− g (cid:20) φR − ω φ g µν ∇ µ φ ∇ ν φ + L ( m ) (cid:21) , where the effective gravitational coupling is proportional to the inverse of the scalarfield φ ( ̺, t ). The constant ω is often called the Brans–Dicke parameter.We would like to understand the dynamics of horizons (black hole and cosmolog-ical) in the Clifton–Mota–Barrow solution. Given that the solution is asymptoticallyFriedman–Lemaˆıtre–Robertson–Walker (FLRW) and dynamical, the most straight-forward way to do so is to look for apparent horizons (note, however, potentialcaveats in the use for apparent horizon, as they depend on the foliation ). An une 8, 2018 8:49 WSPC - Proceedings Trim Size: 9.75in x 6.5in BH˙Vitagliano˙DEF apparent horizon (elsewhere dubbed “trapping horizon” ) is a space- (or time-)likesurface defined as the closure of a surface foliated by marginally trapped surfaces.What follows is based on the analysis and results of Ref. 6.
2. Location of the apparent horizons
The existence and location of the apparent horizons are determined by the condition g rr = 0, where r is the areal radius (not to be confused with the isotropic radius ̺ ). For the Clifton–Mota–Barrow solutions and for an expanding universe with H ≡ ˙ a/a > Hr − ( α − α + 2) α m a ( t ) A ( r ) α − α +1) α − A ( r ) α +1 r = 0 , (1)where H = β/t is the Hubble parameter corresponding to the scale factor a ( t ).The scalar curvature is singular in the limit r →
0, denoting the presence ofa central singularity. Once the variable x ≡ m α̺ has been introduced, eq. (1) canbe solved parametrically for the radius r of the apparent horizon(s) and the timecoordinate t , r ( x ) = a ∗ t β m α (1 + x ) x (cid:18) − x x (cid:19) ( α − α +2) α ,t ( x ) = (cid:26) αm a ∗ β x (1 + x ) α ( α +1) (cid:20) (1 − x ) /α x ( α − α + 2) α (1 − x ) − α − /α (cid:21)(cid:27) β − . As a typical example, we plot the radius of the apparent horizon as a functionof time (in adapted units) for the case ω = 1. The red (dashed) curve is for dust( γ D = 1), the green (solid) curve is for both radiation and stiff matter ( γ R = 4 / γ SM = 2), while the blue (dotted) one corresponds to a cosmological constant( γ Λ = 0). The initial behaviour (a unique, expanding apparent horizon) is commonin all of these different configurations.For dust, radiation, and stiff matter, two further horizons appear at a certaintime t ∗ , the outermost of which expands forever, while the other merges with theinitial horizon, leaving behind a naked singularity covered by a cosmological hori-zon (cf. the contribution by V. Faraoni to these Proceedings, a similar solution byHusain, Martinez, and Nunez, and its interpretation ). The cosmological constantcase is different in the fact that the two horizons appear inside the initial one, asshown in the right panel of Fig. 1.The complete analysis of the Clifton-Mota-Barrow solution reveals a richer phe-nomenology than just the previous example. In particular, the general relativitylimit ω → ∞ turns out to correspond to a generalized McVittie metric. In thiscase an initial naked singularity is then covered by two appearing horizons. This istrue for any equation of state of the cosmological fluid except for the cosmologicalconstant. For the latter the reverse happens, that is, an expanding and a contractinghorizon merge to leave behind a naked singularity. une 8, 2018 8:49 WSPC - Proceedings Trim Size: 9.75in x 6.5in BH˙Vitagliano˙DEF tr tr Fig. 1. Radius of the apparent horizon as function of time in adapted units, for ω = 1. The rightpanel focuses on the early time region of the plot. We conclude with a warning: as already mentioned, apparent horizons dependon the spacetime slicings adopted, namely, specific slicings could imply the absenceof apparent horizons even though the singularity is hidden by an event horizon.From this perspective, the issue of the presence of the naked spacetime singular-ity reported above becomes particularly delicate. Nonetheless, looking for apparenthorizons is probably the most straightforward probe for event horizon and it seemsunlikely that slicing dependence seriously affects the results for a spherically sym-metric foliation of a FLRW spacetime.
Acknowledgements
VV is supported by FCT - Portugal through the grant SFRH/BPD/77678/2011.VF would like to thank NSERC for financial support. TPS acknowledges financialsupport provided under the Marie Curie Career Integration Grant LIMITSOFGR-2011-TPS and the European Union’s FP7 ERC Starting Grant ”Challenging Gen-eral Relativity” CGR2011TPS, grant agreement no. 306425.
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