Sudden Future Singularities and their observational signatures in Modified Gravity
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Sudden Future Singularities and their observational signatures in Modified Gravity a Andreas Lymperis † Department of Physics, University of Patras, 26500 Patras, Greece
ABSTRACT
We verify the existence of Generalized Sudden Future Singularities (GSFS) in quintessence modelswith scalar field potential of the form V ( φ ) ∼ | φ | n where 0 < n < n of the scalar field potential. We then extendthe analysis to the case of scalar tensor quintessence models with the same scalar field potential inthe presence of a perfect fluid, and show that a Sudden Future Singularity (SFS) occurs in this case.We derive both analytically and numerically the strength of the singularity in terms of the power n of the scalar field potential. a Contribution to the School and Workshops on Elementary Particle Physics and Gravity, 2-28 September 2017, Corfu, Greece; to appearin the Proceedings of Science. † [email protected] I. INTRODUCTION
Latest evidence of an accelerating Universe [1–6], has opened new windows in the context of the study of physicsin cosmological scales, and has lead to the consideration of models alternative to ΛCDM . Such models includemodifications of GR (modified Gravity) [7, 8], scalar field dark energy (quintessence) [9, 10], physically motivatedforms of fluids e.g.
Chaplygin gas [11, 12] etc.Some of these dark energy models predict the existence of exotic cosmological singularities, involving divergences ofthe scalar spacetime curvature and/or its derivatives. These singularities can be either geodesically complete [13–16](geodesics continue beyond the singularity and the Universe may remain in existence) or geodesically incomplete[17, 18] (geodesics do not continue beyond the singularity and the Universe ends at the classical level). They appearin various physical theories such as superstrings [19], scalar field quintessence with negative potentials [20], modifiedgravities and others [21, 22].The divergence of the scale factor and/or its derivatives leads to divergence of scalar quantities like the Ricci scalar,thus to different types of singularities or ‘cosmological milestones’ [23, 25, 26]. However geodesics do not necessarilyend at these singularities and if the scale factor remains finite, they are extended beyond these events [22] even thougha diverging impulse may lead to dissociation of all bound systems in the Universe at the time t s of these events[24].Thus, singularities can be classified [27] according to the behaviour of the scale factor a ( t ), and/or its derivativesat the time t s of the event or equivalently, and the energy density and pressure of the content of the universe at thetime t s . A classification of such singularities and their properties is shown in Table I.TABLE I: Classification and properties of cosmological singularities. Name t sing a ( t s ) ρ ( t s ) p ( t s ) ˙ p ( t s ) w ( t s ) T K GeodesicallyBig-Bang (BB) 0 0 ∞ ∞ ∞ finite strong strong incompleteBig-Rip (BR) t s ∞ ∞ ∞ ∞ finite strong strong incompleteBig-Crunch (BC) t s ∞ ∞ ∞ finite strong strong incompleteLittle-Rip (LR) ∞ ∞ ∞ ∞ ∞ finite strong strong incompletePseudo-Rip (PR) ∞ ∞ finite finite finite finite weak weak incompleteSudden Future (SFS) t s a s ρ s ∞ ∞ finite weak weak completeBig-Brake (BBS) t s a s ∞ ∞ finite weak weak completeFinite Sudden Future (FSF) t s a s ∞ ∞ ∞ finite weak strong completeGeneralized Sudden Future (GSFS) t s a s ρ s p s ∞ finite weak strong completeBig-Separation (BS) t s a s ∞ ∞ weak weak completew-singularity (w) t s a s ∞ weak weak complete A particularly interesting type of singularities are the Sudden Future Singularities [21], which involve violation ofthe dominant energy condition ρ ≥ | p | , and divergence of the cosmic pressure of the Ricci Scalar and of the secondtime derivative of the cosmic scale factor Table I. The scale factor can be parametrized as a ( t ) = (cid:18) tt s (cid:19) m ( a s −
1) + 1 − (cid:18) − tt s (cid:19) q , (1.1)where a s is the scale factor at the time t s and 1 < q <
2. For this range of the parameter q , the scale factor and itsfirst derivative, i.e. a, ˙ a respectively, and ρ remain finite at t s . However, the quantities p, ˙ ρ and ¨ a become infinite.Thus, when the first derivative of the scale factor is finite at the singularity, but the second derivative diverges (SFSsingularity [21, 28]), the energy density is finite but the pressure diverges.In the following, we focus on the quintessence models with a perfect fluid, and investigate the strength of the GSFSboth analytically and numerically. We extend the analysis to the case of scalar-tensor quintessence and investigate themodification of the strength of the singularity both analytically (using a proper expansion ansatz) and numerically,by explicitly solving the dynamical cosmological equations. II. THE SETUP
In FRW spacetime with metric ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) (2.1)the most general action involving gravity, nonminimally coupled with a scalar field φ , and a perfect fluid is S = Z (cid:20) F ( φ ) R + 12 g µν φ ; µ φ ; ν − V ( φ ) + L ( fluid ) (cid:21) √− gd x. (2.2)where F ( φ ) is the nonminimal coupling of gravity to the scalar field and L ( fluid ) the fluid term. We have set8 πG = c = 1 and assume spatial flatness ( k = 0). In the case of the scalar-tensor models, corresponding to the action(2.2), we assume a non-minimal coupling linear in the scalar field F ( φ ) = 1 − λφ , even though the results on the typeof the singularity in this class of models are unaffected by the particular choice of the non-minimal coupling.In the special case where the non-minimal coupling F ( φ ) = 1, the action (2.2) reduces to the simple case of quintessecemodels with a perfect fluid S = Z (cid:20) R + 12 g µν φ ; µ φ ; ν − V ( φ ) + L ( fluid ) (cid:21) √− gd x. (2.3)The potential V ( φ ) is of the form V ( φ ) = A | φ | n , A > , (2.4)with 0 < n < A a constant parameter. The dynamical evolution of the scalar field due to the potential is shownin Fig. 1 - - V () FIG. 1: Dynamical evolution of the scalar field potential V ( φ ) = A | φ | n It was shown, through a qualitative analysis [30], that the power law scalar potential (2.4) leads to singularitiesat any scale factor derivative order larger than three, depending on the value of the power n . In particular, for k < n < k + 1, with k >
0, the ( k + 2) th derivative of the scale factor diverges at the singularity. This is in factthe simplest extension of ΛCDM with geodesically complete cosmic singularities and occurs at the time t s , when thescalar field becomes zero ( φ = 0). III. THE QUINTESSENCE CASE
The action in this class of models, is of the form (2.3). The energy density and pressure of the scalar field φ , maybe written as ρ φ = 12 ˙ φ + V ( φ ) and p φ = 12 ˙ φ − V ( φ ) . (3.1)and we assume that the perfect fluid is pressureless ( p m = 0).Variation of the action (2.3) leads to the dynamical equations3 H = 3Ω ,m a + 12 ˙ φ + V ( φ ) (3.2)¨ φ = − H ˙ φ − An | φ | n − Θ( φ ) (3.3)2 ˙ H = − ,m a − ˙ φ (3.4)where a is the scale factor, H = ˙ aa is the Hubble parameter, ρ m = ρ ,m a = ,m a , Ω ,m = 0 . φ ) = ( , φ > − , φ < t → t s i.e. φ →
0, the Hubble parameter H and its first derivative ˙ H remain finite and so does ˙ φ . But in eq. (3.3) there is a divergence of the term φ n − for 0 < n < φ → ∞ as φ →
0. ¨ H also diverges at this point due to the divergence of ¨ φ , as follows by differentiating eq. (3.4). This impliesthat the third derivative of the scale factor diverges, and a GSFS occurs at this point ( i.e. a s , ρ s , p s remain finitebut ˙ p → ∞ ). Thus, the constraints on the power exponents q, r of the diverging terms in the expansion of the scalefactor ( ∼ ( t s − t ) q ) and of the scalar field ( ∼ ( t s − t ) r ) are 2 < q < < r < q to lie in the intervals ( N, N + 1) for N ≥
2, where N ∈ Z + , afinite-time singularity occurs in which d N +1 adt N +1 → ∞ (3.6)but d s adt s → , f or s ≤ N ∈ Z + (3.7)This allows for pressure singularities which are accompanied by divergence of higher time derivatives of the scalefactor (divergence of the fourth-order derivative of the scale factor [31] when p → ∞ ), in Friedmann solutions ofhigher-order gravity ( f ( R )) theories [32].The above qualitative analysis can be extended to a quantitative level by introducing a new ansatz for the scalefactor and the scalar field, containing linear and quadratic terms of ( t s − t ). These terms play an important role, sincethey dominate in the first and second derivative of the scale factor as the singularity is approached.The new ansatz for the scale factor which generalizes (1.1), by introducing linear and quadratic terms in ( t s − t ),is of the form [29] a ( t ) = 1 + ( a s − (cid:18) tt s (cid:19) m + b ( t s − t ) + c ( t s − t ) + d ( t s − t ) q , (3.8)where m = w ) , w the state parameter, b, c, d are real constants to be determined, and 2 < q < a divergesat the GSFS.The corresponding expansion of the scalar field φ ( t ) in the vicinity of the singularity is of the form φ ( t ) = f ( t s − t ) + h ( t s − t ) r (3.9)where 1 < r < φ diverges at the singularity and f, h are real constants to be determined. n = = = - - - ϕ '' ( t ) FIG. 2: Numerical solutions of the second time derivative of the scalar field for n = 0 . , . , .
9. Notice thedivergence at the time of the singularity when the scalar field vanishes.From eq. (3.3) and differentiated eq. (3.4), using the forms of the scale factor (3.8) and the scalar field (3.9), weget two equations that contain only dominant terms in ( t s − t ), in which both the left and right-hand sides diverge atthe singularity for 0 < n <
1, 2 < q < < r <
2. Equating the power laws q and r of the divergent terms weobtain r = n + 1 (3.10) q = r + 1 . (3.11)and it follows that q = n + 2 . (3.12)Figure 2 shows the divergence of the second derivative of the scalar field at the time of the singularity. In figures3a, 3b we plot the numerically verified derived power law dependence (eqs (3.10), (3.12)) of the scalar field and thescale factor respectively, as the singularity is approached. It is clear that eqs (3.10), (3.12) are consistent with thequalitatively expected range of r, q , for 0 < n < t s − t ), in the expression of the scale factor (3.8), play an importantrole in the estimation of the Hubble parameter and its derivative as the singularity is aproached. An interesting resultarises from the derivation of the relation between the coefficients b, c . The relations between these coefficients canlead to relations between the Hubble parameter and its derivative close to the singularity, which in turn correspond n = = = - - - - - ( t s - t ) L o g α ''' ( t ) (a) 3a n = = = - - - - - ( t s - t ) L o g ϕ '' ( t ) (b) 3b FIG. 3: Plots of numerical verification of the q -exponent (3a) and r -exponent (3b) for 3 values of n ( n = 0 . , n = 0 . n = 0 . q and r are identical.to observational predictions, that may be used to identify the presence of these singularities in angular diameter ofluminosity distance data. The relation between b, c is of the form c = ρ ,m a s −
12 ( a s − m ( m − − [( a s − m − b ] a s , (3.13)and thus ˙ H = 3Ω ,m a s − H (3.14)and as a function of redshift parameter z at present time H ( z ) = Ω ,m (1 + z ) [1 − (1 + z ) (1 + z ) − ] + (1 + z ) (1 + z ) − H , (3.15)where H , z are the Hubble and redshift parameter respectively at present time. This result may be used as obser-vational signature of such singularities in this class of models.In the absence of the perfect fluid, the strength of the singularity remains unaffected. This means that the evaluatedrelations of r and q (eqs (3.10), (3.12)) respectively, are exactly the same. The Hubble parameter and its derivativein this case is ˙ H = − H (3.16)and as a function of redshift parameter z at present time H ( z ) = H (1 + z ) (1 + z ) . (3.17)These are the reduced relations of eqs (3.14) and (3.15) respectively, for ρ ,m = 0. IV. MODIFIED GRAVITY: THE SCALAR-TENSOR QUINTESSENCE CASE
The action of the theory, in this class of models, is of the form (2.2). The corresponding dynamical equations are3
F H = 3Ω ,m a + ˙ φ V − H ˙ F (4.1)¨ φ + 3 H ˙ φ − F φ (cid:18) ¨ aa + H (cid:19) + An | φ | ( n − Θ( φ ) = 0 (4.2) − F (cid:18) ¨ aa − H (cid:19) = 3Ω ,m a + ˙ φ + ¨ F − H ˙ F , (4.3)where F φ = dFdφ . From eq. (4.1), it is clear that H, ˙ φ, F, ˙ F all remain finite when φ → t → t s ). However, in eq.(4.2) there is a divergence of the term V φ for 0 < n < φ → ∞ as φ →
0. This means that ¨ F → ∞ because of thegeneration of the second derivative of φ that leads to a divergence of ¨ a in eq. (4.3). Clearly, an SFS singularity (TableI) is expected to occur in scalar-tensor quintessence models, as opposed to the GSFS singularity in the correspondingquintessence models. Thus, the constraints on the power exponents q, r in this case are 1 < q < < r < r and q are q = r (4.4) r = n + 1 , (4.5)which leads to q = n + 1 . (4.6)In figures 4a, 4b we illustrate the numerically verified derived power law dependence eqs (4.5), (4.6) of the scalarfield and the scale factor respectively, as the singularity is approached. Figures 5a, 5b depict the divergence of thesecond derivative, of both the scale factor and the scalar field, at the time of the singularity. n = = = - - - - - ( t s - t ) L o g α '' ( t ) (a) 4a n = = = - - - - ( t s - t ) L o g ϕ '' ( t ) (b) 4b FIG. 4: Numerical verification of the q -exponent (4a) and r -exponent (4b), in the scalar-tensor case, for 3 values of n ( n = 0 . , n = 0 . n = 0 . q and r are identical. n = = = - - - - - - - - α '' ( t ) (a) 5a n = = = - - - - -
10 t ϕ '' ( t ) (b) 5b FIG. 5: Numerical solutions of the second time derivative of the scale factor (5a) and the scalar field (5b) for n = 0 . , . , .
6. Notice the divergence of both the scale factor and scalar field at the time of the singularity.The results (4.5) and (4.6) are consistent with the above qualitative discussion for the expected strength of thesingularity. Thus, in the case of the scalar-tensor theory, we have a stronger singularity at t s , as compared to thesingularity that occurs in quintessence models. This is a general result, valid not only for the coupling constant of theform F = 1 − λφ but also for other forms of F ( φ ) ( e.g. F ∼ φ r ), because the second derivative of F with respect totime, in the dynamical equations, will always generate a second derivative of φ with divergence, leading to a divergenceof ¨ a .The quadratic term of ( t s − t ), in the expression of the scale factor (3.8), is now subdominant as the second derivariveof the scale factor diverges. The only additional term of ( t s − t ) that can play an important role in the estimation ofthe Hubble parameter, is the linear term. Clearly, for the first derivative of (3.8), as t → t s from below, the linear termdominates over all other terms, while the quadratic term is subdominant in the second derivative, in the divergenceof the q -term. Thus, in the case of the scalar-tensor quintessence models H remain finite and dominated by the term b ( t s − t ), while ˙ H → ∞ as t → t s .As in quintessence case of the previous section, in the absence of the perfect fluid, the strength of the singularityremains unaffected. This means that the evaluated relations of r and q , eqs (4.5), (4.6) respectively, are exactly thesame. V. CONCLUSIONS AND DISCUSSION
We have derived analytically and numerically the cosmological solution close to a future-time singularity for bothquintessence and scalar-tensor quintessence models. For quintessence, we have shown that there is a divergence of ... a and a GSFS singularity occurs ( a s , ρ s , p s remain finite but ˙ p → ∞ ) , while in the case of scalar-tensor quintessencemodels there is a divergence of ¨ a and an SFS singularity occurs ( a s , ρ s remain finite but p s → ∞ , ˙ p → ∞ ). In theabsence of the perfect fluid in the dynamical equations, in both cases, we have shown that this result is still valid inour cosmological solution.These are the simplest non-exotic physical models where GSFS and SFS singularities naturally arise. In the caseof scalar-tensor quintessence models, there is a divergence of the scalar curvature R = 6 (cid:16) ¨ aa + ˙ a a (cid:17) → ∞ because ofthe divergence of the second derivative of the scale factor. Thus, a stronger singularity occurs in this class of models.Such divergence of the scalar curvature is not present in the simple quintessence case.We have also shown the important role of the additional linear and quadratic terms of t s − t in the form of the scalefactor as t → t s . However, in the scalar-tensor case the quadratic term becomes subdominant close to the singularity.For quintessence models, we derived relations of the Hubble parameter, H ( z ) = Ω ,m (1 + z ) [1 − (1 + z ) (1 + z ) − ] + (1 + z ) (1 + z ) − H (for the fluid case) and H ( z ) = H (1+ z ) (1+ z ) (for the no fluid case), close to the singularity.These relations may be used as observational signatures of such singularities in this class of models.Interesting extensions of the present analysis include the study of the strength of these singularities in othermodified gravity models e.g. string-inspired gravity, Gauss-Bonnet gravity etc. and the search for signatures of suchsingularities in cosmological luminosity distance and angular diameter distance data. VI. ACKNOWLEDGMENTS
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