Dynamical evolution of nonminimally coupled scalar field in spherically symmetric de Sitter spacetimes
Abstract
We investigate the dynamical behavior of a scalar field nonminimally coupled to Einstein’s tensor and Ricci scalar in geometries of asymptotically de Sitter spacetimes. We show that the quasinormal modes remain unaffected if the scalar field is massless and the black hole is electrically chargeless. In the massive case, the coupling of both parameters produces a region of instability in the spacetime determined by the geometry and field parameters. In the Schwarzschild case, every solution for the equations of motion with \(\ell >0\) has a range of values of the coupling constant that produces unstable modes. The case \(\ell =0\) is the most unstable one, with a threshold value for stability in the coupling. For the charged black hole, the existence of a range of instability in \(\eta \) is strongly related to geometry parameters presenting a region of stability independent of the chosen parameter.
1 Introduction
The evolution of probe fields in black hole backgrounds has long been a very active field of research in theoretical physics [1, 2, 3, and references therein]. Probe field profiles in the time domain present a discrete set of complex frequencies called quasinormal frequencies (QNFs) that can provide valuable information about the structure of spacetime. Each of these frequencies corresponds to a damped vibrational mode of the field, the socalled quasinormal mode (QNM). The set of QNM’s carry specific information about the signature of the geometry (e.g. black hole solutions) and its interaction with fields, since it depends on the parameters that define the metric.
The applications of quasinormal modes are manifold: probing the linear stability of black holes and stars spacetimes [4, 5, 6, 7, 8, 9]; identification of astrophysical black holes through gravitational waves signals [3], experimentally verified by LIGO [10, 11]; studying the role played by such oscillations in the context of gauge/gravity duality, especially in the AdS/CFT [12, 13, 14, 15, 16] and dS/CFT correspondences [17, 18, 19].
The stability of black holes and stars has been discussed in several works [20] since the 50’s with the original paper of Regge and Wheeler analyzing the Schwarzschild singularity [21]. The QNM’s of scalar, Abelian gauge, and fermionic free probe fields evolving in the neighborhood of black holes have also been used to obtain insights about the nature of spacetime. In the case of asymptotically flat black holes these QNM’s are, by the nohair theorems, functions of only the mass M, the electric charge Q, and the angular momentum \(L_\phi \) of the black hole [22]. However, more recently, these theorems were circumvented in the asymptotically AdS black holes and other configurations with nonminimally coupled fields such that hairy black holes solutions have been found [23, 24, 25, 26, 27]. In the latter cases, the QNM’s depend on other hairs of the spacetime, and black hole phase transitions are present.
In AdS/CFT correspondence context, a robust interpretation for the QNM spectra in the view of a quantum field theory at finite temperature (defined at the AdS boundary) is provided: the inverse of the imaginary part of the fundamental quasinormal frequency is understood as a relaxation time of the dual operator at the boundary [13]. Among the applications of AdS/CFT correspondence to condensed matter physics [28, and references therein], we mention the phase transitions at the border theory giving rise to the socalled holographic superconductors [29, 30, 31, 32, 33, 34]: the phenomena yields a specific bulk effect through the QNMs, i.e, growing/decaying oscillations of a given probe field in the bulk correspond to a conductive/superconducting phase at the dual field theory [27]. The presence of instabilities (growing modes) in the quasinormal spectrum therefore indicates a phase transition at the border.
On the dS/CFT correspondence [35], the evolution of probe fields on the gravity side is related to fundamental quantities in the border field theory [17, 18, 19]: the poles of the twopoint correlator of the threedimensional conformal field theory at the boundary scale perfectly the QNMs spectrum of a massive scalar field in the de Sitter spacetime.
Nonminimally coupled (NMC) curvature models were firstly considered in the late 80’s [36], as an alternative gravitation theory. The presence of a scalar field coupled to curvature terms in Einstein–Hilbert action allows for a suitable solution for the inflation exit, and in general has a de Sitter spacetime as the attractor for later times, as should be expected. Besides the traditional terms of NMC models, a few years later, derivative terms were introduced in the action [37], expanding the possibilities for the scalar field potential, characterizing the nonminimally derivative coupling (NMDC) models. From the possible derivative terms, only two significant contributions are in general considered. With a particular scale of the Lagrangian couplings and the cosmological constant, the inflation scenario is generated, as well as the de Sitter spacetime remnant from the curvature equations [38, 39].
The curvature equations coming from NMDC models are of third or higher order, in general. For a particular choice of couplings, however, it is still possible to achieve second order equations: when the Lagrangian derivative terms are placed as Einstein tensor coupled to scalar field components [40]. This choice turns the NMDC into a more suitable (simple) form, as it makes unnecessary to fine tune the scalar field potential.
Beyond the strategic elimination of the fine tuning problem, another possible purpose of the coupling is to perform as a dark matter component, feasible in the form of \(\varLambda \)CDM model [41]. The rate of the scalar field density and total density in the model is slightly different from that of a cold dark matter model, but still in the observationally allowed range. Once NMDC models could be used to describe dark energy and dark matter, they would be instrumental to understand how this coupling affects black holes: for instance, in the context of scalar–tensor gravity exact hairy black hole solutions have been found using NMDC models [24, 25].
In the case of NMDC models, field propagation and quasinormal modes were investigated in a group of papers with a different approach [42, 43, 44]. In [42, 43] the QNMs were obtained in spacetimes with charge, mass, dilaton fields and other hairy geometries. In [44, 45, 46] the dynamical evolution of scalar and vector fields are examined showing the presence of dynamical instabilities associated with a critical value of the NMDC coupling.
Here we are interested in the effect produced on the scalar field equation, given usual black hole geometries as a fixed background. In this approach the probe fields are treated as small perturbations, that are not expected to change the fixed geometry and decay in time. In such case, the corrections of the metric elements are of small order and can be consistently set to zero [1], once the energymomentum tensor for the scalar field is quadratic.
The paper is organized as follows: in Sect. 2 we establish a general equation of motion for the scalar field \(\varPhi \) for spherically symmetric spacetimes. In Sects. 3 and 4 we analyze the dynamical properties of the field in the spacetimes of de Sitter, Schwarzschild–de Sitter, and Reissner–Nordström–de Sitter. In Sect. 5 we present our conclusions and final remarks relative to peculiar features of the coupling for all geometries considered.
2 Equation of motion
3 QNM’s for nonminimally coupled scalar fields evolving in the pure de Sitter and Schwarzschild–de Sitter spacetimes
In this section, we are going to explore the dynamics of nonminimally coupled scalar field in de Sitter and Schwarzschild–de Sitter spacetimes, through the computation of quasinormal frequencies spectrum and modes.
3.1 De Sitter spacetime
We firstly analyze the pure dS case, in which an analytical expression for the scalar QNMs was found in [48], where the probe scalar field is not coupled to the Einstein tensor [\(\eta = 0\) in the Lagrangian (4)].
3.1.1 Purely imaginary frequencies and instabilities
Using the expressions for the frequencies found above, we constraint the values of the NMDC parameter \(\eta \) and the scalar field mass m in order to get purely imaginary QNMs and, more interesting, a range of parameters allowing growing modes, i.e., frequencies with positive imaginary part.
Purely imaginary frequencies have been found in the context of black hole perturbations, and its applications to the AdS/CFT correspondence are manifold. In [49] the authors found a close relation between the Korteweng–de Vries equation and the three dimensional Lifshitz black hole in New Massive Gravity (NMG). They also showed that the scalar QNMs in the hydrodynamic limit are purely imaginary, which in the view of linear response theory corresponds to a solitonic solution. Also in the context of NMG, purely imaginary QNMs were found beyond the hydrodynamic limit in [50]. Furthermore, purely imaginary spectra have been found for a probe scalar field evolving on the geometry of ddimensional Lifshitz black hole [51] and for the Chern–Simmons sector of ddimensional Lovelock black holes [16].
An attempt to give an interpretation of QNMs in the framework of the dS/CFT correspondence [35] was made in [17, 18], where the authors considered the exact QNM spectrum of scalar perturbations on a threedimensional rotating dS black hole and in [19] for a pure \(d\)dimensional dS black hole. In [17, 18], it was found an exact relation between the QNM spectrum and the spectrum of thermal excitations of a Conformal Field Theory, which presents growing modes, leading to regions of instability. Following the same procedure as in [17, 18], it is possible to show that there are growing modes and regions of instability in the case of the 4dimensional dS spacetime with \(\eta \ne 0\).
In short, growing purely imaginary QNMs in the positive branch of the two sets of exact frequencies are present in the spectra, featuring two regions of instability.
3.2 Schwarzschild–de Sitter spacetime
Fundamental quasinormal modes for nonminimally coupled scalar field evolving in Schwarzschild–dS black holes
\(M=\eta = L/6 = m/0.3 = 1\)  \(M=\ell =m/0.5=\eta /2=1\)  \(M=\ell =L/9=m/0.5=1\)  

\(\ell \)  \(\omega \)  L  \(\omega \)  \(\eta \)  \(\omega \) 
0  \(9.6950.04775i\)  5.2  \(0.015900.003730i\)  0  \(0.19820.04479i\) 
1  \(4.9090.04658i\)  5.25  \(0.059270.01367i\)  5  \(0.23520.04397i\) 
2  \(1.9770.04787i\)  5.3  \(0.081770.01876i\)  10  \(0.53810.04349i\) 
3  \(0.53390.04797i\)  5.4  \(0.11310.02565i\)  11  \(0.39070.04344i\) 
4  \(0.43860.04792i\)  6  \(0.20920.04446i\)  50  \(0.19660.05852i\) 
5  \(0.34400.04781i\)  10  \(0.36590.05512i\)  55  \(0.13010.05125i\) 
20  \(0.25060.04757i\)  30  \(0.44690.02382i\)  56  \(0.11050.5731i\) 
50  \(0.16080.04688i\)  50  \(0.45880.01281i\)  500  \(0.11000.05751i\) 
100  \(0.087130.04506i\)  100  \(0.46940.004889i\)  5000  \(0.13180.05088i\) 
Though in first principle, the instability of the spacetime to the scalar field perturbation is not dependent on the multipole number – in the sense that the presence of only one multipole turns the field unstable – numerically, this is not the case. For highly enough \(\eta \) and \(\ell > 0\), the field turns out to be stable no matter the geometry parameters. This is not the case however for \(\ell = 0\), as we may further discuss.
As discussed all along in the literature, the usual evolution of the scalar field after a initial burst in a positive potential is that of a damped oscillator, what characterizes the quasinormal modes. In the pure Schwarzschild–de Sitter case the massive scalar field chooses one of the three different behaviors after the ringing phase: (1) decays exponentially (\(\ell > 0\)), (2) goes to a constant value that scales the cosmological constant (\(\ell = 0\)), (3) oscillates indefinitely as a function of the scalar field mass.
In a more general case, however, a different behavior arises when the potential is not entirely positive between horizons: unstable modes can emerge and the geometry is then expected to change. This is the case for the NMDC \(\eta \) in Schwarzschild–de Sitter geometry we study here: the potential is partly or entirely negative (depending on the coupling and geometry parameters). In this section, we evolve the field for different \(L,\eta \) and \(\ell \). All studied cases take \(L^2 > 27M^2\), which is the causal structure condition for the presence of an encapsulated singularity (by the event horizon) and a cosmological horizon.
In Fig. 1 (right and left panels) we see typical quasinormal mode evolutions for the scalar field for different values of \(\ell \) and L: the higher the multipole number/dS radius, the smaller the frequency of oscillation. The imaginary part of \(\omega \) varies very slowly with l, which is typical for the Schwarzschild–dS geometry also in the absence of couplings, but is majorly affected for the variation of L, diminishing as we increase the cosmological radius. The interesting feature is the emergence of an oscillatory evolution, introduced by the NMDC \(\eta \) for the \(\ell = 0\) mode: there is a quasinormal ringing phase (left panel of Fig. 1) which does not exist in the Schwarzschild–dS case [52], associated now entirely with the renormalized mass of the scalar field.
The effect of the cosmological constant is similar in the Schwarzschild–dS case: the higher the \(\varLambda \), the smaller the real part of \(\omega \). The behavior for \(\omega _I\) is more complicated, oscillating in a given scope of L and becoming arbitrarily small as \(\varLambda \) increases. This can be seen in Fig. 1 (left panel), and in Table 1, which lists quasifrequencies for different values of \(\ell \) and L. In the same table we can also see different \(\omega \) for a range of \(\eta \): the asymptotic values of the coupling are the same as for the massless scalar field propagation in Schwarzschild–dS case.
In the left panel of Fig. 2 we see the transition between stable/unstable dynamics as a function of \(\eta \) for the special case \(\ell =0\). Stable evolution takes place from \(\eta = 0 \) until \(\eta < L^2/3 = 27\), exhibiting the expected decay in time (the potential being only positive). For \(\eta > L^2/3\), on the other hand, the dynamics is always unstable: even for asymptotic \(\eta \), where the potential is partly positive, there is no stable evolution (see right panel of Fig. 2).
Although in the Schwarzschild geometry the instability for \(\eta > \varLambda ^{1}\) is easily verifiable, the situation changes significantly for \(\ell >0\), as it can be seen in the same figure, right panel (\(\ell =1\)), in which the field evolves unstably for \(27< \eta < 31.8\), for the chosen parameters, \(M=\ell =L/9=2m=1\), but decays in time for \(\eta > 31.9\) and the same parameters. Although the fundamental mode destabilizes the geometry from the critical point \(\eta = L^2/3\) on, for the excited modes, there is a second critical value present from which the excited modes are stable. The existence of a point for \(\eta \) from which the field evolves stably is the same found in the charged black hole as we may, see, but differently in the Reissner–Nordström black hole, this fact happens also for \(\ell =0\), thus decreasing the region of instability.
The very special case in which \(\eta = \varLambda ^{1}\) has no solution different from the trivial one, for the massive Klein–Gordon equation, being identically satisfied in the massless case.
Regarding the transition from stable to unstable evolutions, and further again to stable, this transitional behavior is observed also in the Reissner–Nordström geometry, namely, the existence of a region of instability for \(\eta \). We explore the subject in the next section.
Quasiextremal regime
4 QNM’s for nonminimally coupled scalar field evolving in Reissner–Nordström–de Sitter spacetime
 (i)

\(\dfrac{1}{L^2}<\dfrac{p_+(a)}{32M^2}\quad \) and \(\quad a<1\);
 (ii)

\(\dfrac{p_(a)}{32M^2}<\dfrac{1}{L^2}<\dfrac{p_+(a)}{32M^2}\quad \) and \(\quad 1<a<\sqrt{9/8}\),
4.1 Effective potential
Different regions for the potential in RNdS, with \(M=5Q/3=(L/5.4)^2=\ell /2=1\)
Case  \(\eta \)range  Signal of \(V_\mathbb {X} \)  \(r_d\) 

(i)  \(\eta \lesssim 8.46\)  \(V_\mathbb {X}>0\)  \(r_d<r_h\) 
(ii)  \(8.46 \lesssim \eta \lesssim 9.56\)  \(V_\mathbb {X}\gtrless 0\)  \(r_h<r_d<r_c\) 
(iii)  \(9.56 \lesssim \eta \lesssim 9.88\)  \(V_\mathbb {X}<0\)  \(r_c<r_d\) 
(iv)  \(9.88 \lesssim \eta \lesssim 11.52\)  \(V_\mathbb {X}\gtrless 0\)  \(r_c<r_d\) 
(v)  \(11.52 \lesssim \eta \)  \(V_\mathbb {X}>0\)  \(r_c<r_d\) 
Different regions for the potential in RNdS, for \(Q=0.86>Q_c\) (and \(M=\ell /2=(L/6)^2=1\); \(m^2=0\))
Case  \(\eta \)range  Signal of \(V_\mathbb {X}\)  Place of \(r_d\) 

(i)  \(\eta \lesssim 5.84\)  \(V_\mathbb {X}>0\)  \(r_d<r_h\) 
(ii)  \(5.84 \lesssim \eta \lesssim 11.78\)  \(V_\mathbb {X}\gtrless 0\)  \(r_h<r_d<r_c\) 
(iii)  \(11.78 \lesssim \eta \lesssim 12.23\)  \(V_\mathbb {X}<0\)  \(r_c<r_d\) 
(iv)  \(12.23 \lesssim \eta \)  \(V_\mathbb {X}\gtrless 0\)  \(r_c<r_d\) 
The existence of a critical value \(Q_c\) is robust against changes in m and \(\ell \): for every pair \((\ell ,m )\) when \(Q<Q_c\) we can always find a sufficient high \(\eta _k\) such that for any \(\eta > \eta _k\) we have \(V_\mathbb {X}>0\); on the other hand, for \(Q>Q_c\) the potential is strictly negative in \(\mathbb {X}\).
Considering the different character of the potential for the cases (i)–(v), we can investigate the field evolution by obtaining the system’s quasinormal modes and determining whether unstable evolutions are present, or investigate the late time behavior [56] (after the quasinormal ringing). For this reason, we choose to use the characteristic integration over null coordinates to obtain the field profiles together with prony method for the quasifrequencies. For strictly positive gaussianlike potentials, we compare the frequencies to those obtained with WKB method, with good agreement between the results.
4.2 Evolution of scalar field: instabilities and QNM’s
With the acquired quasinormal signal and the prony method [57] we obtain the fundamental quasinormal frequencies up to the critical value of charge \(Q_c \sim 0.85\), as listed in the Table 4.
Fundamental quasinormal modes for nonminimally coupled scalar field evolving in RNdS black holes with different values of Q. The spacetime parameters read \(M=L/6=\ell /2=\eta /50=1\) and \(m^2=0\).The superscript values indicate the deviation of the QNM’s from the RNdS case, \(\frac{\omega _{RN}\omega _{\eta }}{\omega _{RN}}\)
Q  Re \((\omega )\)  Im \((\omega )\)  Q  Re \((\omega )\)  Im \((\omega )\) 

0.05  \(0.2338^{0.0428\%}\)  \(\,0.04905^{0.0204\%}\)  0.50  \(0.2581^{6.55\%}\)  \(\,0.05667^{1.64\%}\) 
0.10  \(0.2346^{0.213\%}\)  \(\,0.04927^{0.0406\%}\)  0.55  \(0.2624^{8.35\%}\)  \(\,0.05849^{2.36\%}\) 
0.15  \(0.2360^{0.466\%}\)  \(\,0.04963^{0.0605\%}\)  0.60  \(0.2667^{10.5\%}\)  \(\,0.06064^{3.36\%}\) 
0.20  \(0.2379^{0.841\%}\)  \(\,0.05015^{0.140\%}\)  0.65  \(0.2710^{13.1\%}\)  \(\,0.06318^{4.75\%}\) 
0.25  \(0.2402^{1.37\%}\)  \(\,0.05081^{0.216\%}\)  0.70  \(0.2751^{16.3\%}\)  \(\,0.06627^{6.68\%}\) 
0.30  \(0.2431^{2.02\%}\)  \(\,0.05163^{0.349\%}\)  0.75  \(0.2792^{20.0\%}\)  \(\,0.07003^{9.25\%}\) 
0.35  \(0.2463^{2.84\%}\)  \(\,0.05261^{0.532\%}\)  0.80  \(0.2833^{24.3\%}\)  \(\,0.07448^{12.7\%}\) 
0.40  \(0.2500^{3.84\%}\)  \(\,0.05377^{0.800\%}\)  0.85  \(0.2824^{29.1\%}\)  \(\,0.07902^{16.6\%}\) 
In Fig. 4 we find two field profiles nearby \(Q \sim Q_c\) (upperright panel) and the instabilities found for high values of Q (lower panels). We can see in the same figure (rightbottom panel) the instability of the near extremal black hole to the scalar field for the overcharged black hole (\(Q>M\)). This is an expected result, given the shape of the potential (very similar to the nearly overcharged black hole, \(Q\sim 0.99M\)) but is not always the case for every \(\eta \): in certain ranges the potential is strictly positive, generating only stable field profiles (e.g. \(M=L/6= \ell / 2 = \eta = 1\)).
The existence of negative regions in the potential does not ensure the presence of instabilities; otherwise, the negativity on \(V_\mathbb {X}\) is related to the presence of an exponential decay in the longtime profile domain. Before \(Q_c\), the field oscillates for very long times (right panel in Fig. 4, \(Q=0.85\)), and beyond this critical charge an exponential decay is shaped as seen in many dSlike geometries [53, 54] (upperright panel in Fig. 4). The exponential decay takes place from \(Q=Q_c\) to another high value of Q, namely \(Q \sim 0.907\) for the assigned parameters (lower panels). For \(Q \gtrsim 0.907\), the field growth is unlimited (Fig. 4 bottomleft pannel). In this case we may not assume the geometry preserves its original shape: it may evolve to a distinct form. In the right panel on the bottom we see the unstable field evolutions for \(M>Q\) to a nearextreme (overcharged) black hole with \(\eta \): in every case, the field grows indefinitely showing an unstable behavior.
The presence of a transitional behavior seems to occur also for the variation of \(\eta \): by taking fixed M, Q and L we investigate the presence of quasinormal modes and instabilities in regions (i)–(v). In Fig. 5 we see different profiles for a large range of \(\eta \).
Fundamental quasinormal modes for nonminimally coupled scalar field evolving in RNdS black holes with varying \(\eta \). The spacetime parameters read \(M=L/7=\ell =2Q=1\) and \(m^2=0\)
\(\eta \)  Re \((\omega )\)  Im \((\omega )\)  \(\eta \)  Re\((\omega )\)  Im \((\omega )\) 

0  0.2034  \(\,0.07374\)  13.29  0.4258  \(\,0.03163\) 
3  0.2067  \(\,0.07308\)  13.2935  0.4389  \(\,0.01148\) 
7  0.2159  \(\,0.07213\)  19  0.1407  \(\,0.08782\) 
10  0.2328  \(\,0.07074\)  21  0.1524  \(\,0.08197\) 
11  0.2449  \(\,0.07008\)  60  0.1835  \(\,0.07739\) 
13  0.3287  \(\,0.06457\)  5000  0.1884  \(\,0.07569\) 
13.15  0.3553  \(\,0.06061\)  \(10^9\)  0.1884  \(\,0.07568\) 
13.28  0.4115  \(\,0.04171\) 
In Table 5 it is make clear the influence of varying\(\eta \) in the scalar field propagation: for the first region of the potential, the higher the \(\eta \), the higher the quality factor of the black hole.^{1} We must be attentive still, of the high variation when getting closer to the frontier of (i) in \(\eta \): from \(\eta = 13.29\) to \(\eta = 13.2935\) we have a \(\varDelta \eta \rightarrow 0.026\%\) variation whereas \(\varDelta \omega _{R} \rightarrow 3.1\%\) and \(\varDelta \omega _{I} \rightarrow 64\%\). This type of change characterizes a variation similar to that occurred in the near extremal regime (when the accretion of small amounts of charge in the black hole induces huge variations in the spectra of the oscillation).
Another interesting picture in the quasinormal spectrum with NMDC is the existence of an asymptotic value of \(\omega \) for high \(\eta \): in the table we can see, to the 4 figure, the QNM is the same for \(\eta = 5000\) and \(\eta = 10^9\), both cases in region (v). The last feature we emphasize, is the highest values of Im\((\omega )\) and the smallest for Re\((\omega )\) both in region (iv). This is an expected feature in relation to the imaginary part, as long as regions (ii), (iii) and (iv) are the unstable ones.
Critical values of \(\eta \) (the field evolves stably after \(\eta > \eta _c\))
\(M=2Q=L/7=1\) and \(m^2=0\)  

\(\ell \)  0  1  2  3  4  5  10  20 
\(\eta _c\)  \(16.25^{\pm 0.05}\)  \(17.55^{\pm 0.05}\)  \(18.25^{\pm 0.05}\)  \(18.75^{\pm 0.05}\)  \(19.05^{\pm 0.05}\)  \(19.35^{\pm 0.05}\)  \(20.05^{\pm 0.05}\)  \(20.35^{\pm 0.05}\) 
From the same table, we realize that \(\eta _c\) increases for increasing \(\ell \). Possibly the values of \(\eta _c\) approach a finite asymptote when we take \(\ell \rightarrow \infty \), given the growing of the \(\ell \) x \(\eta _c\) curve, what is not possible to be investigated numerically.^{2}
As in the de Sitter geometries with black holes [52] the scalar field multipole \(\ell =0\) is a special case. Although not conclusive, for late times, the field tends to increase very slowly to a constant value (for very late times). For instance, taking \(M=Q/2=\eta /500=L/7=1\), the evolution seems to evolve very slowly to an asymptotic value (\(\tilde{R}\sim 0.03\)), for late times.
In general, for all multipole number, we demonstrate the presence of a gap of instability in \(\eta \)range for the scalar field: when \(\eta < 13.2935\) or for \(\eta > \eta _c\) the field evolves stably, being unstable if \(\eta < \eta _c\), in regions (iii) and (iv) (as stated before, it is not possible to obtain numerical integration in region (ii)).
In the regime of high cosmological radius, we can see the formation of region (iii) and (iv) in the potential when \(\eta > \frac{L^2}{3}\), but no region (v) as a general feature. Even for small values of charge, we have no region (v), but the gap for existence of region (iii) is very small in \(\eta \). As an example, let us assume a geometry with \(M=100Q=\ell =1\), \(m =0\) and \(L=6 \times 10^{5}\). The region (iii) for negative potential almost vanishes: it endures a range of \(\varDelta \eta \sim 10^{10}\), after the critical \(\eta \rightarrow \varLambda ^{1}\) (for \(Q=1/2\) and the same parameters, \(\varDelta \eta \sim 10^{6}\)). The region after that, region (iv) appears for every \(\eta \). Region (v) will only emerge in cases with very small black hole charges (\(Q<Q_c \sim 10^{5}\)), for example nearby \(\eta \sim 4\varLambda ^{1}\). The general behavior remains, however: at some point for each geometry, we will have the critical value of \(\eta \) from which the field evolves stably.
The situation changes drastically, though, if we add a small scalar field mass to the last scenario. Taking \(m \sim 0.1\), as an example, we have a range \(\varDelta \eta \sim 10^9\) of unstable fields after the point \(\eta ^{(2)} \sim 1.2 \times 10^9\). In Fig. 5 in lowerright panel we see the coupling of \(\eta \) and the scalar field: in general, the higher the scalar field mass, the higher the value of \(\eta \) for the formation of a stable region of oscillations in the potential.
The potential implies unstable scalar evolution, what can be seen by analyzing the extra term, \(2f/r^2\): it is always positive in the region of field propagation. In such case, the wave equation is the same as that for the scalar field in Reissner–Nordström–de Sitter geometry, with a negative term inside the brackets. Even though this term varies with r, the fact that it is always negative is sufficient to assure the unstable evolution of the scalar field whenever \(\ell = 0\): the field propagating in RNdS geometry with negative square mass (even of very small masses) is unstable (the same result being true in our case). This result is very similar to that of the Schwarzschild section: the instability for \(\eta > L^2/3\), comes as a result of the effective mass being negative in that limit.
The numerical data obtained by evolving the scalar filed in the potential (42) turns out unstable in all tested parameters, for \(M=1\), \(\ell = 0\), \(L=7\) and \(L=50\), and \(Q=0.01, 0.1, 1.001\) (as expected).
Quasiextremal regime
The quasiextremal regime in a RNdS black hole has two possible horizon coalescence, \(r_y=r_h\) (high Q) or \(r_c=r_h\) (high \(\varLambda \)). In the first case, given the high values of charges, region (v) never exists. In this case, all the tested profiles of region (iii) and (iv) for \(\ell \) come out stable whenever \(\ell >0\). On the other hand, taking for example, \(M=L/6=1\) and \(\delta \equiv \frac{Q_{ext}Q}{Q_{ext}} \sim 10^{9}\), for asymptotic \(\eta \), the scalar field turns out unstable. The potential forms region (iii) for \(11.7<\eta < 12.3\), but, as long as all field profiles evolve unstably in region (v), \(\eta > 11.7\) represents an unstable range of parameter. This was tested for multiple \(\eta \) and \(\ell =1\), but can be also take as granted for other \(\ell >1\) as long as the deep of the potential grows in those cases. Again we have most probably a stable evolution for \(\ell =0\), qualitatively similar to the one discussed in the previous subsection for the nonextremal case. In that way we can still assure the presence of region (i) in the potential when \(\eta < 2.22\), and the field evolves stably as a quasioscillation or an exponential decay after the initial burst.
When the cosmological constant is high, we have a more interesting frame. If we take, for instance, \(M=5Q/3=1\) and \(L=4.8587\) (\(\varLambda = 0.999998\varLambda _{ext}\)), regions (ii) and (iv) happen for very small intervals in \(\eta \) of order of \(10^{4}\). In such case, the evidence of a gap of instability is very pronounced. For \(\eta < 7.49\) or \(\eta > 8.29\), the field evolves stably for every \(\ell \).^{3}
5 Final remarks
In the present work we discussed the effect of a nonminimally derivative coupling on the dynamics of a scalar field propagating in asymptotically dS spacetimes. Three different cases were studied: the de Sitter, Schwarzschild–de Sitter, and Reissner–Nordström–de Sitter metrics.
Considering the evolution of a scalar probe field in a fourdimensional dS spacetime, we computed the quasinormal spectrum when the NMDC term \(\eta \) is present. We found growing quasinormal modes in the positive branch of frequencies leading to regions of instability. In the context of dS/CFT correspondence, we generalize the result for the twopoint Hadammad function, showing that its poles match with the regions of instability in the quasinormal spectrum.
In the case of Schwarzschild–de Sitter geometry, the presence of an \(\eta \)term in the field equation also introduces instabilities in the quasinormal spectra for a given range of \(\eta \). The effect of the coupling is to modify the square mass of the scalar field turning it negative in certain ranges of values, presenting expected instabilities for the field evolution (\(\ell =0\)).
Surprisingly, the massless scalar field equation is not affected by the coupling. The spectra of frequencies is stable, as expressed by the usual Schwarzschild–de Sitter quasinormal modes. This was shown to be the case also in the de Sitter spacetime.
The same is not true for the Reissner–Nordström black hole in a dS geometry, where even the massless scalar field is affected for the nonminimally coupling constant. The potential is significantly more complicated, compared to the chargeless case, possessing five qualitative different regions according to its sign. In regions (iii) and (iv) we have two critical constants, \(\eta ^{(2)}\) and \(\eta ^{(3)}\), determined by the spacetime parameters, such that for \(\eta ^{(2)}< \eta < \eta ^{(3)}\) unstable modes are present. The range of \(\eta \) for which unstable modes are present grows as we increase the charge of the black hole.
A range of instability for \(\eta \) occurs for every \(\ell \) (differently from [45, 46]). The frequencies are sensitive to the variation of the \(\eta \)parameter, being the quasinormal spectrum particularly affected by its presence.
For every \(\eta \) it is always possible to find a range of charges of the black hole for which unstable modes are present, suggesting \(\eta \) might be an appropriate order parameter for studying critical phenomena in these systems.
In the quasiextremal limit for Schwarzschild–de Sitter and Reissner–Nordström–de Sitter, the quasinormal spectra was obtained exactly, following the approach of [53, 54], and the observed behavior is similar to that of the nonquasiextremal case.
The investigation of the presence of instabilities is a fruitful field of research. In this work, the peculiar evolution of a probe scalar field in a number of geometries revealed critical phenomena which may be related to second order phase transitions present in the corresponding CFT side of theory. Nonminimally coupled models enable a vast amount of dynamical field analysis, with parameter ranges over which the spacetime is unstable being a particularly important feature.
Footnotes
 1.
Re\((\omega )\) increases and Im\((\omega )\) decreases.
 2.
The higher the \(\ell \), the higher the time (in the field) to which we must integrate in order to obtain the exponential growth/decay of the field. This represents a geometrical growth in time of computation versus an arithmetic growth in \(\ell \).
 3.
Regions (ii) and (iv) form around (7.4895, 7.4921) and (8.285, 8.289), respectively.
Notes
Acknowledgements
The authors would like thank Jefferson Stafusa Elias Portela for critical comments to the manuscript. This work was supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais), Brazil.
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