Modified dispersion relations and a potential explanation of the EDGES anomaly.

The Experiment to Detect the Global Epoch of Reionisation Signature (EDGES) collaboration has recently reported an important result related to the absorption signal in the Cosmic Microwave Background radiation spectrum. This signal corresponds to the red-shifted 21-cm line at z ≃ 17.2 , whose amplitude is about twice the expected value. This represents a deviation of approximately 3.8 σ from the predictions of the standard model of cosmology, i.e. the Λ CDM model. This opens a window for testing new physics beyond both the standard model of particle physics and the Λ CDM model. In this work, we explore the possibility of explaining the EDGES anomaly in terms of modified dispersion relations. The latter are typically induced in unified theories and theories of quantum gravity, such as String/M-theories and Loop Quantum Gravity. These modified dispersion relations affect the density of states per unit volume and thus the thermal spectrum of the Cosmic Microwave Background photons. The temperature of the 21-cm brightness temperature is modified accordingly giving a potential explanation of the EDGES anomaly.

Introduction

Predictions of General Relativity (GR) have been tested with high accuracy ranging from the solar system to the cosmological scales. Despite this success, GR is an incomplete theory at short distance and time scales (for example, near black holes and cosmological singularities), and perhaps at large distances as well, where dark components and/or modifications of GR are invoked to explain the accelerated phase of the present Universe. It is expected that the inconsistencies at small scales can be resolved within the framework of quantum gravity (QG), which incorporates the principles of GR and quantum theory, and provides a description of the microstructure of space-time at the Planck scale.

Among the various attempts towards formulating a theory of QG, String/M-theory and Loop Quantum Gravity (LQG) remain as important candidates. A consequence of these theories is that space-time appears non-commutative (NC) at the fundamental level [1-5], and in some situations, may also exhibit a varying speed of light [6, 7]. This gives rise to non-local field theories and a modification of the dispersion relation of the quantum fields in a NC space-time. For example, one of the consequences of String Theory (as well as of M-Theory) is space-time non-commutativity [1], with the latter leading to modified dispersion relations [8]. Related to this is the fact that owing to quantum fluctuations, the usual canonical commutation relation also gets modified [9-14] (see also Refs. [15-18]). It must be pointed out however, that there are several other approaches to QG that also predict the existence of a minimum measurable length, which in turn represents a natural cutoff and induces a departure from the relativistic dispersion relation. These approaches include space-time foam models [19-21], spin-network in LQG [22], space-time discreteness [23], spontaneous symmetry breaking of Lorentz invariance in string field theory [24] or in NC geometry [25], Horava’s approach [26, 27], and Doubly Special Relativity (DSR) [21, 28, 29]. In Ref. [30], the authors proposed an extension of DSR to include curvature, also known as Doubly General Relativity, in such a way that the geometry of space-time does depend on the energy E of the particle used to probe it (gravity’s rainbow) [21]. The general form of the modified dispersion relation (MDR) reads [31]where the (rainbow) functions and depend on the Planck energy (for details see for instance Refs. [32-39]). Now, whenever , i.e., one deviates from the standard relativistic dispersion relation, as we shall show below, the Planck radiation spectrum changes as well. This in turn may be able to explain the anomaly, which the Experiment to Detect the Global Epoch of Reionisation Signature (EDGES) collaboration has recently reported [40]. In the range , the EDGES collaboration found an anomalous absorption profile, with a brightness temperature minimum at , which has a magnitude of about a factor of two greater than predicted by the CDM model. It is this anomaly that we propose to explain using MDRs in this work. It turns out that the standard MDRs do not adequately explain the EDGES anomaly. However, by imposing redshift dependent MDR parameters, or by imposing a non-trivial power dependence for the MDRs, we are able to provide a viable explanation for the EDGES anomaly. A non-trivial power dependence of a MDR is also discussed in Ref. [41].

The rest of the paper is organized as follows. In the next section, we briefly review some of the important special cases of the above MDR. Following this, in Sect. 3, we estimate the parameters in the models that we consider from the results of the EDGES experiment. Finally, we summarize our results and conclude in Sect. 4.

MDR and modification of thermal spectrum

As stated in the Introduction, MDR is predicted by various theories of QG, and has the most general form of Eq. (1). The rainbow functions can in the most general case be expressed in a power series expansion (MacLaurin series) as and , where constraints and must be imposed to obtain the standard relativistic dispersion relation at low energies. Here we consider some of the interesting special cases.Here are dimensionless parameters, with and to be the linear and quadratic GUP parameters, respectively. It is often assumed that , so that the modifications of the dispersion relations are non-negligible at Planck scales. However, one may relax such a restriction and investigate signals of new physics at a new intermediate scale . Such a length (energy) scale cannot exceed the well-tested electroweak length scale, , so the consequent upper bound .

Case 1: , , which is one of the most studied in literature. Here is a parameter which signifies the effective scale of the modification, and is the order of the modification. A complete theory of QG should fix both of them. However, in this work we study the modifications for different values of and , and in particular, we consider three special cases. The first case is compatible with LQG and NC space-time [42, 43], while the next two are compatible with the linear and quadratic Generalized Uncertainty Principle (GUP) respectively [44, 45]: where restrictions on from Ref. [44] have been relaxed to include both positive and negative values. In general, and, specifically, in the presence of a strong gravitational field , where is the 00 component of the metric [44, 45]. However, in the dark ages, most of the hydrogen gas was in a very weak field, and, therefore, we can set , as far as space-time curvature corrections to the MDR are concerned.

and ,

and ,

and ,

Case 2: , proposed for explaining the spectra from GRBs at cosmological distances [19].

Case 3: , , with . The case has been proposed for models in which a varying speed of light occurs [7]. The case has been proposed in Refs. [21, 46]. For , , and , we recover Case 1. GUP provides another case for this form of [45]:

, and ,  .

The MDR given by Eq. (1), for the case of photons readsso that, following Refs. [7, 46], one may derive the modified thermal spectrum .

The density of states per volume for photons (which have 2 polarization states) is written asBy considering the MDR in Eq. (2) and using , we obtain the following density of stateswhere the two ‘speeds’ in the above equation turn out to bewhere . Therefore, we can write the modified density of states asThe modified thermal spectrum is then obtained using1, where is the Bose-Einstein distribution, is the inverse temperature and is the Boltzmann constant. The modified thermal spectrum then reads aswhereis the standard thermal distribution of photons and R is the correction factor, formally defined in the following section. Note that the standard result from Eq. (8) is obtained from Eq. (7) when the MDR parameters vanish, i.e., .

Experimental bounds

In this section we study the effects of the modified thermal spectrum given by Eq. (7), induced by the MDR given in Eq. (2), on the 21-cm cosmology. Details of 21-cm cosmology are given in Appendix A. This is related to the history of the universe, and represents a new framework for probing fundamental physics [49] (for other models see Refs. [50-59]). In particular, we focus on the recent release of the EDGES collaboration [40] (see also Ref. [60]).

EDGES High and Low band antennas probe the frequency ranges 90–200 MHz and 50–100 MHz, respectively, overall measuring the 21-cm signal within the redshift range , corresponding to an age of the Universe , i.e., the dark ages. This includes the epochs of reionization and cosmic dawn, in which the first astrophysical sources form. At , the observed magnitude of the absorption line2 is about a factor of two greater than the one predicted by the CDM model. At the redshift of the minimum of the 21-cm line, i.e., , and frequency of CMB radiation, i.e., , one has a 21-cm brightness temperature (, including estimates of systematic uncertainties). Since at one has , Eq. (A5) implies [49, 60]. Moreover, in the context of the CDM model, one also getsandwhere and are the redshift and the temperature at the time when the gas and radiation decouple. Using (A5), one infers . Notice that the minimum is saturated for , which corresponds to . As a consequence of these results, one finds that the best fit value for is times lower than expected within the CDM.

The 21-cm CMB photons absorbed at fall clearly in the Rayleigh–Jeans tail since , where is the hyperfine transition energy of the hydrogen atom. The energy density of the photons, i.e., Eq. (8), evaluated at , readswhere . Only photons with energy at could be absorbed by the neutral hydrogen producing a 21-cm absorption global signal. For explaining the EDGES results, we consider the given by Eq. (7). Therefore, we define the parameter R to study the discrepancy from the CDM model aswith and defined in Eqs. (7) and (11), respectively. It may appear that such a modification may affect the optical depth (introduced in Appendix A) and, therefore, the intensity and shape of the 21-cm line profile. However, as shown in Appendix B, such a modification does not affect in any way. The experimental values from the EDGES experiment can then be explained by imposing (see Ref. [60] for details)Parameter R in Eq. (12) is then only a function of F, and E, since everything else except the relevant correction cancels out. Since we can in general write the rainbow functions f and g as a power series in , we can also write the function as a power series expansionNote that , which corresponds to the standard CDM result. The parameter R from Eq. (12) for such a general expression readsEither Eq. (12) or Eq. (15) above can be used to estimate R for the cases studied here, compare with experimentally measured values and obtain bounds on the various parameters.

Case 1: , . The ratio R reads for arbitrary parameters and . We take a look at the special cases:

For and , we have The ratio R is plotted as a function of in Fig. 1. To fit the EDGES experimental bounds, the parameter is fixed at .

Fig. 1

R vs for fixed energy eV

For and we have The ratio R is plotted as a function of for both branches in Fig. 2. However, only the branch with can fix . To fit the EDGES experimental bounds, the parameter is fixed at .

Fig. 2

R vs for fixed energy eV. The branch is presented in dash-dot blue and the branch is presented in solid black

For and we have The ratio R is plotted as a function of in Fig. 3. To fit the EDGES experimental bounds, the parameter is fixed at .

Fig. 3

R vs for fixed energy eV

Case 2: , . The ratio R reads The ratio R is plotted as a function of in Fig. 4. To fit the EDGES experimental bounds, the parameter is fixed at .

Fig. 4

R vs for fixed energy eV

Case 3: , . The ratio R reads for arbitrary parameters , and . We take a look at the special case:

For , and , we have The ratio R is plotted as a function of for both branches in Fig. 5. However, only the branch with can fix . To fit the EDGES experimental bounds, the parameter is fixed at .

Fig. 5

R vs for fixed energy eV. The branch is presented in dash-dot blue and the branch is presented in solid black

At this point it should be stressed that the above plots indicate that the MDRs provided by cases 1, 2 and 3, give at redshift . These values are much larger than the bound set by the electroweak scale . To verify the compatibility with known observations and obtain the bounds on the above parameters in the current epoch (), we compare the experimental precision of the CMB temperature [61] (see also Refs. [62, 63]) of a perfect black body to the theoretical deviation due to MDRs in the current epochIn the above, R(E) is given by Eq. (12) and is given in terms of the CMB temperature in the current epoch. We obtain Eq. (23) by expressing from . The parameters in the current epoch then must satisfy an upper bound of to be consistent with the observed CMB spectrum in the current epoch. The bound obtained from the electroweak experiments is stronger than that, so it should be used as the relevant MDR parameter bound in the current epoch. The EDGES anomaly at combined with the above bound at suggest that the above parameters should be increasing functions of the redshift z. Therefore, we also expect R to increase with z for a given energy E and have a value of at .

The compatibility of such MDRs with epochs earlier than should be taken into consideration as well. For example, in the epoch of the Big Bang Nucleosynthesis (BBN), at [64], a bound of was obtained in [65] for the quadratic GUP parameter , which corresponds to an upper bound for the MDR parameters. Therefore, the values of the MDR parameters, measured by the EDGES anomaly are consistent with the BBN measurements, even if they increase to at . This supports the increasing trend of the redshift dependence of the MDR parameters and may in fact provide a clue in determining the exact form of this dependence. Estimations of the MDR parameters from the modified CMB spectrum would not be relevant in the BBN epoch, since it has not been created until the epoch of recombination at [64].

The standard MDRs used in this work can be found in Refs. [21, 42–46] as mentioned in Sect. 2, but they consider the MDR parameters as constants. The assumption that the MDR parameters are functions of another parameter, such as redshift, is fairly new. However, such an assumption is indirectly supported by Ref. [66], where the author finds a mass/radius dependent GUP parameter. This is also supported by the difference between estimations of the quadratic GUP parameter in tabletop experiments, where [67-71], and astrophysical/cosmological observations, where [66, 72–76]. This shows that the MDR parameters can in fact be dependent on scale or redshift.

Since the usual models of modified dispersion relations can not explain the EDGES anomaly, it is also legitimate to investigate if it can be explained by considering the cases analyzed here in which , namely they are fixed to the electroweak scale, while and are treated as free parameters. We only consider cases 1 and 3, since case 2 has no other parameters to tweak. Also, we did not separately consider the special case 1, iii), because it is automatically considered as .

In Fig. 6, we plot R from Eq. (21) vs for fixed and for fixed . The values of R for fall outside the EDGES bounds and cannot provide an explanation for the EDGES anomaly. However, the values of R for fall inside the EDGES bounds twice in a narrow range of around and can therefore provide an explanation for the EDGES anomaly. Changing the parameter only moves the peak to a different location.

Fig. 6

R vs for fixed energy eV. The branch is presented in dash-dot blue and the branch is presented in solid black

At this point a number of comments are in order. First, the power dependencies on and of these cases are shown in Figs. 7 and 8, respectively. It is easily seen that the EDGES anomaly can be explained by powers and . We also notice that we can only set an upper bound to the powers and , since the electroweak length scale is an upper bound for the new length scale. Second, the stringent values for and , to resolve the EDGES anomaly, with their respective errors will be available in the future, when the true new length scale will be estimated and known with higher energy accelerators and astrophysical observations. Third, we also point out that power , which means that the correction decreases with increasing E as also seen in case 1. It may be noted that negative and positive is equivalent to positive and negative to leading order.

Fig. 7

R vs for fixed and energy eV. The solid black, the dash-dot blue and dotted red lines represent cases 1) i) and ii) (positive and negative branch) respectively

Fig. 8

R vs for fixed and energy eV. The dash-dot blue, solid black and dotted red lines represent respectively

Conclusion

In this work, we have studied the possibility that MDRs can account for the recent results of the EDGES collaboration, which has discovered an anomalous absorption signal in the CMB radiation spectrum. This signal is larger by about a factor of 2 with respect to the expected value (assuming that the background is described by the CDM model), i.e., the EDGES anomaly. In particular, we have shown that the most commonly considered MDRs, namely cases 1-3, lead to a modified thermal spectrum and to the subsequent estimation of the parameters . Unfortunately, the parameter values at redshift are outside the bounds allowed by, e.g., the electroweak experiments, since . However, given the precision of the CMB temperature in the current epoch, , we were able to constrain these parameters to an upper bound to be consistent with the observed CMB black body spectrum. The estimation of the MDR parameters from the EDGES anomaly at , the bound obtained from electroweak experiments at and the BBN bound at suggest that the MDR parameters should be functions of redshift z and as such could explain the EDGES anomaly. We can assume that the evolution of MDR parameters with time in the current epoch is slow or nearly constant, since we observe the same physics in all observable astrophysical objects such as distant galaxies. However, the time evolution of MDR parameters could have been faster in the early stages of the Universe as the EDGES anomaly suggests.

There is also another way out! As seen in Figs. 6, 7 and 8 and explained there, letting the powers , and vary does also explain the anomaly for finite ranges of those powers. To precisely fit the EDGES experiment, and set , bounded by the electroweak scale, we have studied the possibility of treating the powers , and of the MDRs as free parameters and estimating upper bounds to their values. Similar results were found in Ref. [41]. However, MDRs with non-trivial power dependencies require further research to better understand their importance for QG theories.

The results in this work indicate that MDRs originating from existing theories and thought experiments with minimal measurable length can provide a mechanism which explains the EDGES anomaly only if the MDR parameters are increasing functions of redshift z. Also, if the true QG theory with minimum measurable length predicts non-trivial deformation parameters as obtained from Figs. 6, 7 and 8, then such a theory can also provide a viable mechanism to explain the EDGES anomaly as well. It will be interesting to study the consequences of such deformation parameters in various contexts, such as GRBs physics [19]. We hope to report on this in the future.