Extreme parameter sensitivity in quasidilaton massive gravity
We reanalyze the behavior of Friedmann-Lemaître-Robertson-Walker cosmologies in the recently proposed quasidilaton massive-gravity model, and discover that the background dynamics present hitherto unreported features that require unexpected fine-tuning of the additional fundamental parameters of the theory for an observationally consistent background cosmology. We also identify new allowed regions in the parameters space and exclude some of the previously considered ones. The evolution of the mass of gravitational waves reveals non-trivial behavior, exhibiting a mass-squared that may be negative in the past, and that presently, while positive, is larger than the square of the Hubble parameter, . These properties of the gravity-wave mass have the potential to lead to observational tests of the theory. While quasidilaton massive gravity is known to have issues with stability at short distances, the current analysis is a first step toward the investigation of the more stable extended quasidilaton massive gravity theory, with some expectation that both the fine-tuning of parameters and the interesting behavior of the gravity-wave mass will persist.
The standard cosmological model, CDM, describes the acceleration of the universe by properly adjusting the cosmological constant . While this simple model is consistent with current observational data, other models provide alternative explanations of this acceleration. For example, some models attribute the acceleration to the presence of a dynamical component known as dark energy Creminelli et al. (2009); Zlatev et al. (1999); Chiba et al. (2000), and others to a modification of the gravitational laws on cosmological distances Hamed et al. (2004); Nojiri and Odintsov (2006); Sotiriou and Faraoni (2010); Dvali et al. (2000); Maartens (2004). The questions will be to what extent it is possible to discriminate among the different models from observations, and whether any of the models are better at fitting the data than what is currently the most parsimonious explanation, .
The next generation of experiments (such as EUCLID euc () or DESI des ()) will provide an unprecedented amount of observational data. However, there is now a wide range of candidate theories. For instance, different modifications of general relativity primarily in the infrared have been considered by many authors (see Clifton et al. (2012) for a recent review), and probably still others have yet to be proposed. Ultimately, the predictions of each candidate model must be confronted with data. This includes not just cosmological data but data on all scales where the models make calculable predictions that can be tested observationally or experimentally.
Within one interesting class of theories, the current acceleration era is associated to the presence of a mass term for the graviton (for a historical overview, motivations and an updated description of different proposed massive gravity theories, see de Rham (2014); Tolley (2015)). Here we consider a particular modification of general relativity known as quasidilaton massive gravity (QDMG), which we summarize in the Section II. This theory was proposed in D’Amico et al. (2013a), as an extension of the dRGT theory of massive gravity de Rham and Gabadadze (2010); de Rham et al. (2011), and contains an additional scalar degree of freedom: the quasidilaton. The main motivation for such extension is the absence of isotropic and homogeneous cosmological background solutions in dRGT D’Amico et al. (2011). Indeed, it has been shown that QDMG has solutions with spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background metrics D’Amico et al. (2013a). Moreover, it has been found that (even in the absence of a cosmological constant) there are solutions for which at late times the metric approaches to a de Sitter metric, providing a plausible (self-accelerating) explanation of the accelerated expansion of the universe D’Amico et al. (2013a); Gannouji et al. (2013); Bamba et al. (2014). The quasidilaton theory has three parameters more than CDM.
In this paper we will perform a careful analysis of the background cosmological evolution, taking into account the main goal of describing the observed expansion history of the universe. While other authors have made preliminary investigations Gannouji et al. (2013); Bamba et al. (2014) of the background evolution in QDMG, a more detailed reexamination reveals important new insights. The allowed set of parameters split into two disconnected regions characterized by “low” and “high” values of a dimensionless parameter of the theory, (which multiplies the kinetic term of the quasidilaton). In the region with low values of , while viable background solutions exist for a wide range of values of the Lagrangian parameter nominally called the graviton mass , with , a careful fine-tuning of the dimensionless constants and is required. The permitted values of , and thus describe a very thin 2-dim surface in the parameter space. In the other region, the parameter is constrained to be much smaller than , and the larger it is, the narrower the 2-dim surface of allowed and .
The paper is organized as follows. After summarizing the theory QDMG in Section II, in Section III we present the dynamical equations. In Section IV we analyze the existence of viable de Sitter fixed-point attractors. By exploring the 4-dimensional parameter space of the theory, in Section V, we assess the viability of a self-accelerating explanation of the current expansion of the universe. An important outcome of our analysis is that, in order to reproduce an expansion history consistent with data, the graviton mass parameter must also be fine-tuned to a value that depends on other parameters of the model.
In Section VI we study the evolution of the mass of gravitational waves for the allowed set of parameters. We find the current value of to be generically larger than the current Hubble constant even when we set the graviton mass parameter . In the past (for example at redshifts relevant for the Cosmic Microwave Background) can be either real or imaginary. For a conservative choice of , in the past, with at last scattering. While this precludes the development of a catastrophic instability when is imaginary, nevertheless potentially there could be observable cosmological signatures. These merit further investigation Dubovsky et al. (2010); Emir Gümrükçüoǧlu et al. (2012); Bessada and Miranda (2009).
Ii Theory of quasidilaton massive gravity
We consider the action for the quasidilaton theory D’Amico et al. (2013a):
where is the Planck mass and, in addition to the Einstein-Hilbert action , a contribution characterizes the quasidilaton scalar field . In addition to the quasidilaton kinetic term, includes three interaction terms: Here
with square brackets denoting a trace, and
The non-dynanmical “fiducial metric” is built from four Stückelberg fields (),
In the space of Stückelberg fields, the theory enjoys the Poincare symmetry D’Amico et al. (2013a)
and in addition, there is a global symmetry given by
with an arbitrary constant.
The addition of to the action, introduces four new parameters: the dimensionless kinetic coupling , the graviton mass parameter , and the coupling constants and . As shown below, the cosmological solution depends sensitively on the values of these parameters.
Iii The background cosmological equations
We consider a spatially flat Friedmann-Lemaître-Robertson-Walker ansatz, for which
The fiducial metric reduces to
The minisuperspace action for the background metric and fields can now be written as
and we have defined
Varying the action with respect to leads to
where , with
We use time reparameterization freedom to set .
In summary, the independent background equations are:
the constraint equation (21), or its integral
the Friedman equation,
and we have included the contributions of matter and radiation;
the conservation of the stress-energy tensor obtained from
where a prime means derivative with respect to ;
the conservation of the stress-energy tensors of matter, and of radiation .
(Note that using the constraint equation (21) one can show that the equation obtained by taking the variation of with respect to is not an independent equation.)
Iv De Sitter fixed point analysis
We start by investigating the future background evolution of the quasidilaton massive gravity model. The CDM concordance model predicts the universe will approach a de Sitter phase in the future. Though we do not know the future of the universe, we require our model to reproduce this prediction, consistent with recent practice Gannouji et al. (2013); Bamba et al. (2014).
We rewrite eq. (24) in terms of the relative energy densities
Employing eq. (21) and assuming that we obtain
Eq. (23) implies that as , . Therefore the set of variables , , , will approach constants in the asymptotic future. We study the dynamical stability of the system by means of the following equations:
can be obtained by differentiating the first Friedmann equation (24) to obtain
As noted above, we focus on de Sitter fixed points, and require that these critical points are attractors. The de Sitter critical points relative to the system (34), (35) and (IV) are given in Table 1, where
To assess the stability we compute the matrix form of the perturbation equations linearized around each of the fixed points. Then, the linear asymptotic stability of each fixed point can be studied by analyzing the signs of the eigenvalues of that matrix. If the sign of the real part of every eigenvalues is negative, then the critical point is an attractor. The results are shown in the last column of Table 1.
This analysis indicates there are three possible late-time de Sitter fixed points, , and . For each, constraints on the parameters and are obtained by requiring that and . For the point we obtain and ; for , and ; for , . Noticing that one must insist that , we found different constraints than Gannouji et al. (2013); Bamba et al. (2014).
Consider more closely the fixed point B. Given the and constraints for , we obtain . The constraint equation (23) implies that in the asymptotic future . Moreover (23) requires that should be unbounded either above or below in order to have a past history. is a polynomial in X, so this is impossible if 111Notice that and cannot be crossed in the past history. as it is. Thus cannot be a well-defined fixed point. Recalling that , a similar argument can be applied to the point .
V Cosmological evolution and parameter fixing
The aim of this section is to study the evolution of the relevant background quantities in agreement with the results of the previous section and with the observed cosmological history. That depends on the initial conditions, on the expansion history and on the fixed point . By means of this analysis we constrain the four parameters of quasidilaton massive gravity: .
Given that we are dealing with the background energy density evolution, we can consider neutrinos to be relativistic, since the value of is not negligible only in the radiation era when neutrinos were indeed relativistic. The spectrum of the CMB today is precisely measured, so we accurately determine . For relativistic neutrinos, is proportional to . Therefore we assume is known and we fix it by .
To be definite we also fix , close to the best fit value Planck Collaboration et al. (2015a). That corresponds in CDM to . We stress that this choice will not qualitatively affect our conclusions on the quasidilaton massive gravity background evolution.
To fix the initial conditions we require , where and are given by (31), (32). In this way we obtain a 9-th order polynomial that has no analytical solutions. In Fig. 1 we plot the function for . The red points represent the values of and where the initial conditions are satified. From eq. (23) we obtain . Finally, notice that is unbounded as ; this implies that for our model the correct past evolution of the background is allowed only if .
The dark energy equation-of-state parameter is constrained by observations. To compute for the QDMG model we first define the total effective equation-of-state parameter
We must require that in agreement with the current limits Planck Collaboration et al. (2015b).
The quasidilaton massive gravity model shows a particular feature – scales as matter at early times Gannouji et al. (2013). Indeed, from the analysis above we have and . At early times . Therefore we find 222To be consistent with observations, or in the past, so or .
It follows that, for redshifts , would contribute to the effective matter energy density! Therefore should be negligible in the radiation era in order to have a viable expansion history. We demand that .
In order to identify the allowed ranges of the four parameters of QDMG, the main computational obstacle is to find the solutions of the initial condition, namely . In principle could have from to allowed solutions for each value of the QDMG parameters. However, after enforcing all the observational conditions, we find that there is never more than viable solution. We identify two disconnected allowed regions in the -dimensional space of parameters, one shows just low- values (hereafter REG1) and the other one high- values (hereafter REG2).
In Fig. 2 and 3 we present the constraints for the (REG1) parameter space. After marginalizing over we find that is constrained to as we report in Fig. 2. On the other hand, marginalizing over , the contour plot reported in Fig. 3 shows that the values are tightly related to the value . Once we know two of the three parameters, the other one is determined to a good approximation. In other words the quasidilaton massive gravity theory presents a fine-tuning of the parameters.
Repeating the procedure for (REG2) we find a different behavior. In Fig. 4 we see that while the allowed interval depends on and it becomes larger as decreases. The region is again dependent, however the dependence now is different than for (REG1) as we show in Fig. 5. For (REG2) if then and becomes effectively a function of , so we find a fine-tuning of parameters. On the other hand if the graviton mass is small, i.e. , the other parameters are no longer strongly constrained.
In our analysis we did not compute the whole expansion history for each point in the four dimensional parameter space for practical computational reasons. As an illustrative example, we choose two set of allowed parameters for (REG1) and (REG2) and we plot the evolution of the energy densities in Fig. 6. As expected, the two panels, are consistent with the observed expansion history.
The parameter fine-tuning we found practically reduces from four to three the effective parameters of the quasidilaton massive gravity theory. We expect that studying the perturbations will further constrain the theory. Some of those perturbations will be unstable.
Vi Gravity waves
In this section we focus on the evolution of the mass of the gravitational waves. We consider tensor perturbations around the background metric solutions,
with , and . After a straightforward calculation, one gets the quadratic Lagrangian for
where the mass of the gravitational waves is given by 333Note that this expression coincides with the one computed for the extended quasidilaton in Heisenberg (2015) .
We start by computing the ratio at redshift , relevant for CMB, for the two disconnected regions (REG1) and (REG2) defined in the previous section. An exploration of the values computed reveals there is a minimum and maximum for each of the regions. The results are presented in Table 2. We see that a real mass as large as can be obtained, even for our conservative choice . For both parameter regions (REG1) and (REG2), we note the mass can be imaginary. However, the maximum absolute values turn out to be much smaller than the Hubble rate, preventing the development of a full instability. It is worth noting that so far signatures in the Cosmic Microwave Background (CMB) due to a non-vanishing have been studied assuming this mass is always real Dubovsky et al. (2010); Emir Gümrükçüoǧlu et al. (2012); Bessada and Miranda (2009). Our results suggest that one should explore also the possibility of having cosmological gravitational waves with a small but imaginary mass at the relevant redshifts for CMB.
In Fig. 7 we plotted the evolution of the ratio for the parameters given in Table 2. We notice that at low redshifts (and in particular at ) the mass becomes positive, and is larger than , despite the fact that . In order to assess the generality of this result we computed the ratio at varying the parameters in the two disconnected allowed regions of the -dimensional space, and we obtained its maximum and minimum value. The results are shown in Table 3. We see that is larger than , even for values of , and it is up to a factor of larger than for . The existence of a minimum value of that is larger than is remarkable, since this represents a motivated observational threshold. That is, if one could constrain to be smaller than one would be able to rule out a self-accelerating explanation of the current acceleration of the universe within the QDMG theory. It would be interesting to see whether an analogous result holds for other theories that also aim to provide a self-accelerating explanation. Unfortunately, current experiments are still far from probing Olive et al. (2014). Moreover, the upper limits one can obtain are in general model dependent, since they are based on assumptions involving different scales of the theory. This represents a challenge for both theory and observations, and highlights the ongoing importance of working out predictions within the framework of specific models of modified gravity.
|Max. Min.||Max. Min.|
|Max. Min.||Max. Min.|
The combination of General Relativity and the Standard Model of particle physics are demonstrably and remarkably successful descriptions of the world on scales up to and including the solar system. On larger scales, there is a need either to modify the theory of gravity or to introduce new forms of dark matter and dark energy. The most parsimonious solution would be to identify candidates for the latter in the Standard Model, and such candidates may exist for dark matter (see for example Witten (1984); Lynn et al. (1990)) and evade existing constraints Jacobs et al. (2015), although the phenomenological successes of MOND (see for example McGaugh (2015)) cannot be entirely dismissed as an indication of the need to modify gravity on galactic scales. For the observed cosmic acceleration, the situation is even less clear. A cosmological constant is the canonical explanation, but despite decades of attempts has as yet no clear explanation in the Standard Model. The need for observational probes of possible dark energy and modified gravity explanations is thus paramount.
One possibility would be to develop some general phenomenological classification of possible deviations of gravity from GR. The Parametrized post-Newtonian approach is one such program, in the context of almost-Schwarzschild backgrounds. Such generic approaches have also been attempted in the cosmological context (eg. Lue et al. (2004); Hu and Sawicki (2007)). However, in the context of a highly non-linear theory such as GR, the observational consequences of small theoretical departures from GR can be quite ideosyncratic. While phenomenological parametrization of observables may be convenient, and even useful, they may not capture (or may capture poorly) the specific phenomena or behaviour that result from actual models. Careful examination of specific individual models can therefore be both instructive and essential.
In this paper, we have studied the (homogeneous) cosmological solutions of quasidilaton massive gravity. A study of the linear perturbations around the asymptotic self-accelerated cosmological solution of this theory (which corresponds to a De Sitter background metric) has been done in D’Amico et al. (2013b); Emir Gumrukcuoglu et al. (2013). These studies have revealed that the kinetic term of one of the scalar perturbations becomes negative for short wavelengths, indicating that the theory may have a ghost instability that shows up at short distances. This is indeed the case at linear level. Several authors De Felice and Mukohyama (2014); De Felice et al. (2013) have therefore extended the theory by allowing for a new coupling, which can be properly adjusted to make the scalar sector stable at linear level. This extended quasidilaton massive gravity theory (EQDMG), has been considered by other authors Motohashi and Hu (2014); Kahniashvili et al. (2015); Heisenberg (2015).
Although this current reconsideration of the background cosmological solutions of QDMG was performed as a first step for a full analysis of the EQDMG, it revealed important attributes of the QDMG cosmology, which we expect to carry over qualitatively or in detail to EQDMG. The first is that observationally viable QDMG cosmologies require fine-tuning of parameters. In particular, the allowed values of the graviton mass parameter, is a tightly constrained function of the coupling constants and , with only a very narrow tolerance around a central value . This fine-tuning, and the precise value of , is dictated by observational constraints on the dark energy properties.
The second observation is that some small (but possibly non-negligible) fraction of what manifests as (i.e. ) today, was (i.e. ) in the past. The transition from one equation of state to the other was sudden and probably not well-captured by a linear parametrization of . The expected difference between at high redshift (as measured in the CMB) and at low redshift (as measured, say in large scale structure), could be the source of recently noted tensions in different determinations of Planck Collaboration et al. (2015c); Delubac et al. (2015). While the details of these behaviors of the background cosmomology are likely to be altered in EQDMG, it is plausible that these qualitative features are robust.
We have also analyzed the phenomenology of the graviton. The governing equations for the graviton mass (which is not equal to the graviton mass parameter ) are the same in QDMG and EQDMG. We therefore expect to gain useful insights for the extended model provided that the background solutions do not depend sensitively on the new parameter of the extended theory. We find that the graviton mass-squared typically is negative at redshifts well above , indicating an instability. This includes redshifts where such physics may well imprint itself on the CMB. At any given time , so we do not expect the instability to lead to many e-foldings of growth. Nevertheless, if this persists in EQDMG, it may be another opportunity to see evidence of modified gravity in CMB observations.
Regarding vector perturbations, according to equations (4.16) and (4.17) of Heisenberg (2015), the square of the speed of propagation, , can be recast as , where the absence of ghost instability is guaranteed provided 444We note this inequality is satisfed in the QDMG case for parameters in the allowed regions. Notice in particular that when becomes negative, the absence of ghost instability implies becomes also negative. Therefore, we expect that a detail analysis of the perturbations will further reduce the region of allowed parameters.
In a future work, we will therefore extend our analysis to the EQDMG theory, taking into account the constrains from the study of the perturbations, anticipating hopefully that these observable effects will indeed persist.
Acknowledgements.We thank C. de Rham for helpful comments, and G. Gabadadze, M. Fasiello, D. Müller and A. Tolley for discussions. DLN is thankful the Case Western Reserve University for hospitality when this work was initiated. We acknowledge the use of the xAct - xPand package for Mathematica Pitrou et al. (2013); Martín-García (). SA and GDS are supported by a Department of Energy grant DE-SC0009946 to the particle astrophysics theory group at CWRU.
- Creminelli et al. (2009) P. Creminelli, G. D’Amico, J. Norena, and F. Vernizzi, JCAP 0902, 018 (2009), arXiv:0811.0827 [astro-ph] .
- Zlatev et al. (1999) I. Zlatev, L. Wang, and P. J. Steinhardt, Physical Review Letters 82, 896 (1999), astro-ph/9807002 .
- Chiba et al. (2000) T. Chiba, T. Okabe, and M. Yamaguchi, Phys. Rev. D 62, 023511 (2000), astro-ph/9912463 .
- Hamed et al. (2004) N. A. Hamed, H. S. Cheng, M. A. Luty, and S. Mukohyama, Journal of High Energy Physics 5, 074 (2004), hep-th/0312099 .
- Nojiri and Odintsov (2006) S. Nojiri and S. D. Odintsov, eConf C0602061, 06 (2006), arXiv:hep-th/0601213 [hep-th] .
- Sotiriou and Faraoni (2010) T. P. Sotiriou and V. Faraoni, Reviews of Modern Physics 82, 451 (2010), arXiv:0805.1726 [gr-qc] .
- Dvali et al. (2000) G. Dvali, G. Gabadadze, and M. Porrati, Physics Letters B 485, 208 (2000), hep-th/0005016 .
- Maartens (2004) R. Maartens, Living Reviews in Relativity 7, 7 (2004), gr-qc/0312059 .
- (9) http://sci.esa.int/euclid/ .
- (10) http://desi.lbl.gov/ .
- Clifton et al. (2012) T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys.Rept. 513, 1 (2012), arXiv:1106.2476 [astro-ph.CO] .
- de Rham (2014) C. de Rham, Living Reviews in Relativity 17, 7 (2014), arXiv:1401.4173 [hep-th] .
- Tolley (2015) A. J. Tolley, Lect.Notes Phys. 892, 203 (2015).
- D’Amico et al. (2013a) G. D’Amico, G. Gabadadze, L. Hui, and D. Pirtskhalava, Phys. Rev. D 87, 064037 (2013a), arXiv:1206.4253 [hep-th] .
- de Rham and Gabadadze (2010) C. de Rham and G. Gabadadze, Phys. Rev. D 82, 044020 (2010), arXiv:1007.0443 [hep-th] .
- de Rham et al. (2011) C. de Rham, G. Gabadadze, and A. J. Tolley, Physical Review Letters 106, 231101 (2011), arXiv:1011.1232 [hep-th] .
- D’Amico et al. (2011) G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava, and A. J. Tolley, Phys. Rev. D 84, 124046 (2011).
- Gannouji et al. (2013) R. Gannouji, M. W. Hossain, M. Sami, and E. N. Saridakis, Phys. Rev. D 87, 123536 (2013), arXiv:1304.5095 [gr-qc] .
- Bamba et al. (2014) K. Bamba, M. W. Hossain, R. Myrzakulov, S. Nojiri, and M. Sami, Phys. Rev. D 89, 083518 (2014), arXiv:1309.6413 [hep-th] .
- Dubovsky et al. (2010) S. Dubovsky, R. Flauger, A. Starobinsky, and I. Tkachev, Phys. Rev. D 81, 023523 (2010).
- Emir Gümrükçüoǧlu et al. (2012) A. Emir Gümrükçüoǧlu, S. Kuroyanagi, C. Lin, S. Mukohyama, and N. Tanahashi, Classical and Quantum Gravity 29, 235026 (2012), arXiv:1208.5975 [hep-th] .
- Bessada and Miranda (2009) D. Bessada and O. D. Miranda, JCAP 8, 033 (2009), arXiv:0908.1360 [astro-ph.CO] .
- (23) Notice that and cannot be crossed in the past history.
- Planck Collaboration et al. (2015a) Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al., ArXiv e-prints (2015a), arXiv:1502.01589 .
- Planck Collaboration et al. (2015b) Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, and et al., ArXiv e-prints (2015b), arXiv:1502.01590 .
- (26) To be consistent with observations, or in the past, so or .
- (27) Note that this expression coincides with the one computed for the extended quasidilaton in Heisenberg (2015) .
- Olive et al. (2014) K. Olive et al. (Particle Data Group), Chin.Phys. C38, 090001 (2014).
- Witten (1984) E. Witten, Phys. Rev. D 30, 272 (1984).
- Lynn et al. (1990) B. W. Lynn, A. E. Nelson, and N. Tetradis, Nuclear Physics B 345, 186 (1990).
- Jacobs et al. (2015) D. M. Jacobs, G. D. Starkman, and B. W. Lynn, MNRAS 450, 3418 (2015).
- McGaugh (2015) S. S. McGaugh, Canadian Journal of Physics 93, 250 (2015), arXiv:1404.7525 .
- Lue et al. (2004) A. Lue, R. Scoccimarro, and G. Starkman, Phys. Rev. D 69, 044005 (2004), astro-ph/0307034 .
- Hu and Sawicki (2007) W. Hu and I. Sawicki, Phys. Rev. D 76, 104043 (2007), arXiv:0708.1190 .
- D’Amico et al. (2013b) G. D’Amico, G. Gabadadze, L. Hui, and D. Pirtskhalava, Classical and Quantum Gravity 30, 184005 (2013b), arXiv:1304.0723 [hep-th] .
- Emir Gumrukcuoglu et al. (2013) A. Emir Gumrukcuoglu, K. Hinterbichler, C. Lin, S. Mukohyama, and M. Trodden, ArXiv e-prints (2013), arXiv:1304.0449 [hep-th] .
- De Felice and Mukohyama (2014) A. De Felice and S. Mukohyama, Physics Letters B 728, 622 (2014), arXiv:1306.5502 [hep-th] .
- De Felice et al. (2013) A. De Felice, A. E. Gümrükçüoğlu, and S. Mukohyama, Phys. Rev. D 88, 124006 (2013), arXiv:1309.3162 [hep-th] .
- Motohashi and Hu (2014) H. Motohashi and W. Hu, Phys. Rev. D 90, 104008 (2014), arXiv:1408.4813 [hep-th] .
- Kahniashvili et al. (2015) T. Kahniashvili, A. Kar, G. Lavrelashvili, N. Agarwal, L. Heisenberg, and A. Kosowsky, Phys. Rev. D 91, 041301 (2015), arXiv:1412.4300 .
- Heisenberg (2015) L. Heisenberg, JCAP 4, 010 (2015), arXiv:1501.07796 [hep-th] .
- Planck Collaboration et al. (2015c) Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al., ArXiv e-prints (2015c), arXiv:1502.01597 .
- Delubac et al. (2015) T. Delubac, J. E. Bautista, N. G. Busca, J. Rich, D. Kirkby, S. Bailey, A. Font-Ribera, A. Slosar, K.-G. Lee, M. M. Pieri, J.-C. Hamilton, É. Aubourg, M. Blomqvist, J. Bovy, J. Brinkmann, W. Carithers, K. S. Dawson, D. J. Eisenstein, S. G. A. Gontcho, J.-P. Kneib, J.-M. Le Goff, D. Margala, J. Miralda-Escudé, A. D. Myers, R. C. Nichol, P. Noterdaeme, R. O’Connell, M. D. Olmstead, N. Palanque-Delabrouille, I. Pâris, P. Petitjean, N. P. Ross, G. Rossi, D. J. Schlegel, D. P. Schneider, D. H. Weinberg, C. Yèche, and D. G. York, Astron. Astrophys. 574, A59 (2015), arXiv:1404.1801 .
- (44) We note this inequality is satisfed in the QDMG case for parameters in the allowed regions.
- Pitrou et al. (2013) C. Pitrou, X. Roy, and O. Umeh, Classical and Quantum Gravity 30, 165002 (2013), arXiv:1302.6174 [astro-ph.CO] .
- (46) J. M. Martín-García, http://www.xact.es .