Consistent equivalence principle tests with fast radio bursts
MMNRAS , 1–6 (2020) Preprint 24 February 2021 Compiled using MNRAS L A TEX style file v3.0
Consistent equivalence principle tests with fast radio bursts
Robert Reischke (cid:63) , Ste ff en Hagstotz † and Robert Lilow ‡ Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB),German Centre for Cosmological Lensing, 44780 Bochum, Germany The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Roslagstullsbacken 21A, SE-106 91 Stockholm, Sweden Department of Physics, Technion, Haifa 3200003, Israel
24 February 2021
ABSTRACT
Fast radio bursts (FRBs) are astrophysical transients of still debated origin. So far severalhundred events have been detected, mostly at extragalactic distances, and this number is ex-pected to grow significantly over the next years. The radio signals from the burst experiencedispersion as they travel through the free electrons along the line-of-sight characterised bythe dispersion measure (DM) of the radio pulse. In addition, each photon also experiencesa gravitational Shapiro time delay while travelling through the potentials generated by thelarge-scale structure. If the equivalence principle (EP) holds, the Shapiro delay is the same forphotons of all frequencies. In case the EP is broken, one would expect an additional dispersionto occur which could be either positive or negative for individual sources. Here we suggest touse angular statistics of the DM fluctuations to put constraints on the EP parametrized by thepost-Newtonian parameter γ . Previous studies su ff er from the problem that the gravitationalpotential responsible for the delay diverges in a cosmological setting, which our approachavoids. We carry out a forecast for a population of FRBs observable within the next years andshow that any significant detection of the DM angular power spectrum will place constraintson the EP that are by a few orders of magnitude more stringent than current limits. Key words: cosmology: theory, large-scale structure of Universe, radio continuum: transients
Fast radio bursts (FRBs) are short transients lasting usually onlya few milliseconds with frequencies ranging from ∼
100 MHz toseveral GHz. Due to the free electrons in the ionized intergalacticmedium (IGM), the pulse experiences a dispersion ∆ t ∝ ν − , wherethe amplitude is called the dispersion measure (DM) (e.g. Thorntonet al. 2013; Petro ff et al. 2015; Connor et al. 2016; Champion et al.2016; Chatterjee et al. 2017) and is proportional to the integratedelectron density along the line-of-sight. While the mechanism forthe radio emission is still under debate, their isotropic occurrenceand large observed DM suggest an extragalactic origin, so that theDM can be used to test the distribution of di ff use electrons in thelarge-scale structure (LSS). Several authors therefore proposed touse the DM inferred from FRBs as a cosmological probe, using ei-ther the averaged signal (Zhou et al. 2014; Walters et al. 2018) orthe statistics of DM fluctuations (Masui & Sigurdson 2015; Shi-rasaki et al. 2017; Rafiei-Ravandi et al. 2020; Reischke et al. 2021;Bhattacharya et al. 2020; Takahashi et al. 2021). (cid:63) E-mail: [email protected] † E-mail: ste ff [email protected] ‡ E-mail: [email protected] For a compilation of currently proposed mechanisms for FRBs, seehttps: // frbtheorycat.org (Platts et al. 2019). Recently, FRBs have also been employed to test the equiv-alence principle (EP), which is one of the key axioms of Gen-eral Relativity. If the EP holds, photons of di ff erent frequencies,as well as any other freely falling massless particles, should followthe same null-geodesic. Measuring any deviation from this wouldimmediately indicate EP-breaking physics beyond the current stan-dard model. There is a number of papers (e.g. Krauss & Tremaine1988; Wei et al. 2015, 2016a,b; Wu et al. 2016, 2017; Yang et al.2017; Yu et al. 2018; Xing et al. 2019; Yao et al. 2020) dealing withthis type of measurements, which all rely on measuring the Shapirotime delay (Shapiro 1964) for di ff erent messengers. The most re-cent ones focus on gravitational waves, gamma-ray bursts or FRBs.It is key to all Shapiro delay tests that the transient is short com-pared to the associated Shapiro delay, since this in principle deter-mines the sensitivity of the measurement. All of the measurementslisted above assume a metric with weak perturbations that vanish atinfinity. In the parametrized post-Newtonian (PPN) formalism, thisleads to the well known Shapiro delay equation: t grav = − + γ c (cid:90) r o r e d λ U (cid:0) r ( λ ) (cid:1) , (1)where the integral is performed from the source at r e to the ob-server at r o . We introduced the PPN parameter γ measuring devi-ations from the Newtonian time delay (Will 2014), U denotes the © a r X i v : . [ a s t r o - ph . C O ] F e b Reischke, Hagstotz and Lilow
Newtonian gravitational potential, and c is the speed of light. ForGeneral Relativity one finds γ =
1, independent of frequency orparticle type. Deviations from the EP can be parametrized in termsof the di ff erence ∆ γ between the γ values of di ff erent messengers.Wei et al. (2015), for example, used FRBs to constrain ∆ γ betweenphotons of di ff erent frequencies.Minazzoli et al. (2019) pointed out that the standard formula-tion is not well defined in a cosmological setting. Since the startingpoint is a perturbed Minkowski metric, eq. (1) can only be appliedto nearby sources. For corrections in a cosmological setting, alsosee Nusser (2016). In addition, since the cosmic density field doesnot vanish at infinity, the resulting naive Shapiro delay in eq. (1)diverges. It is usually argued that only taking into account contri-butions from some structures along the line-of-sight (and neglect-ing the remaining cosmic structure) leads to conservative limits, butMinazzoli et al. (2019) demonstrate that the neglected contributionscan be positive or negative. Thus, any partial reconstruction of thedensity field is insu ffi cient.In this paper we demonstrate how FRBs can be used to setlimits on EP breaking similar to the procedure suggested by Nusser(2016), but by using statistical observables of the DM. This avoidsdealing with unphysical quantities and the problematic boundaryconditions for a cosmological setting. If the EP is violated, the ob-served DM consists of a term arising due to the dispersion inducedby the free electrons in the LSS structure and and by a second termarising from the fact that the photons at two di ff erent frequenciesexperience a di ff erent Shapiro delay. Since FRBs are very short,any breaking of the EP would lead to a very strong deviation inthe time delay due to the long paths involved in a cosmological set-ting. Therefore one can expect that any measurement of statistics ofthe DM at the expected level immediately rules out any substantialdeviation from ∆ γ = Λ -cold-dark-matter ( Λ CDM) scenario withfiducial parameters given by the best fit values of Aghanim et al.(2020).
The observed time delay, ∆ t obs , between di ff erent frequency bandsof an astrophysical transient can be split into several contributions: ∆ t obs = ∆ t int + ∆ t LIV + ∆ t m + ∆ t grav . (2) ∆ t int is the intrinsic time delay due to the source and the type oftransient. In the case of FRBs this can be split into the DM con-tribution ∆ t DM and a potential source contribution ∆ t s . The secondterm, ∆ t LIV , describes Lorentz invariance violation and is ignoredhere. The term ∆ t m is a potential additional dispersion in case thephoton is massive. From the relativistic dispersion relation E = c p + m γ c (3)one finds that massive photons of di ff erent frequencies propagateat di ff erent speeds. This also introduces a delay in the arrival time between di ff erent frequencies of the FRB signal: ∆ t m ( z ) = (cid:32) m γ c π (cid:126) (cid:33) A ( z ) (cid:16) ν − − ν − (cid:17) , (4)with the redshift evolution given by A ( z ) = (cid:90) d z (cid:48) + z (cid:48) ) H ( z (cid:48) ) . (5)Note that the frequency dependence has the same shape as the dis-persion caused by the intergalactic plasma, so for any single sourcethis would contribute to the total DM.Both ∆ t LIV and ∆ t m produce non-zero e ff ects on the cosmolog-ical background. As the current study will be only concerned withperturbations, these two contributions are ignored in what follows.With this we are left with ∆ t obs = ∆ t s + ∆ t DM + ∆ t grav , (6)where the last term is the di ff erence in the gravitational time delay,eq. (1), between photons of di ff erent frequency bands. The general problems with EP tests based on the Shapiro delayhave already been pointed out by Minazzoli et al. (2019), but wewill briefly recap them here for the case of FRBs. So far, most ofthe cosmological tests consider a localised FRB (or any other objectsuitable for a di ff erential Shapiro delay measurement as outlined insection 1) and use the reconstructed potential, for example the oneof the Milky Way, to calculate the predicted Shapiro delay. Sincethe Shapiro delay is usually much larger than the delay induced bythe dispersion of the free electrons along the line-of-sight, any EPbreaking ( ∆ γ (cid:44)
0) would completely dominate the time delay. It isthen assumed that this gives a conservative bound on ∆ γ since theShapiro delay will just increase when adding more reconstructeddata, i.e. another cluster along the line-of-sight.The problem with this procedure is that the gravitational po-tential along the line-of-sight is influenced by all particles in theUniverse. Thus, the Shapiro delay diverges if the density does notfall o ff . The reason for this problem is the choice of coordinates as-sociated with eq. (1). At spatial infinity the potential and its deriva-tive are expected to vanish. This boundary condition might be suit-able for isolated objects, but it is not fulfilled in the cosmologicalsetting with constant background density. In particular, Minazzoliet al. (2019) show that the cosmological part completely dominatesthe naive expectation of the Shapiro delay from any individual Ke-plerian potential, e.g. of the Milky Way.Lastly, Minazzoli et al. (2019) show how to renormalise thedivergence by choosing an appropriate time coordinate and derivean expression for the Shapiro delay which, however, is no longermonotonic with respect to the number of sources used to recon-struct the potential along the line-of-sight. Therefore one cannotplace a conservative bound on ∆ γ , at least on cosmological scales,by just considering a subset of sources for which a reconstructedpotential along the line-of-sight is readily available. In a cosmological setting, we can avoid the problem of a di-verging time delay by considering a weakly perturbed Friedman-Robertson-Walker (FRW) line element in conformal Newtonian
MNRAS000
MNRAS000 , 1–6 (2020) quivalence principle tests with FRBs gauge within the PPN formulation:d s = − (cid:18) + φ c (cid:19) c dt + a ( t ) (cid:18) − γφ c (cid:19) d x (7)with the gauge potential φ , the scale factor a and the comovingcoordinates x . The corresponding time delay between photons ofdi ff erent frequencies can then be written as (Nusser 2016) ∆ t grav ( ˆ x ) = ∆ γ c (cid:90) χ s d χ a ( χ ) φ (cid:0) ˆ x χ, a ( χ ) (cid:1) , (8)where χ is the comoving distance at the background level. This ex-pression does not diverge in a cosmological setting since it respectsthe cosmological symmetry assumptions by construction. Whilethis looks very similar to eq. (1), the perturbation φ is a Gaussianrandom field with zero mean, and therefore the time delay can ac-quire positive and negative contributions along the line-of-sight.Note that this expression is not suited for individual nearbylines-of-sight dominated by objects that cannot be described assmall, linear perturbations to a FRW background. Furthermore, theabsolute value of φ in any finite volume is not an observable quan-tity. However, these problems can be avoided by observing angu-lar correlations of time delays. They are insensitive to individualnearby objects, and are independent of monopole contributions andthus the absolute value of φ . The fact that local fluctuations of ∆ t grav can be negative poses no problem since any EP breaking changesthe expected correlations. Angular correlations can therefore beused to place an upper bound on the strength of EP violation. The observed time delay in direction ˆ x for a source at redshift z isinterpreted as a total DM via ∆ t obs ( ˆ x , z ) ∝ DM tot ( ˆ x , z ) ν − . (9)More explicitly, the time delay between two measured frequencies ν , is ∆ t obs ( ˆ x , z ) = t ν , obs ( ˆ x , z ) − t ν , obs ( ˆ x , z ) = K DM tot ( ˆ x , z ) (cid:16) ν − − ν − (cid:17) , (10)where we absorb all the constants in K = e / (2 π m e c ). Here, e and m e denote the charge and mass of an electron, respectively. TheDM and time delays can therefore be used interchangeably. We willmostly work with the DM from now on since it is the quantity ofinterest for FRB correlation studies.Any additional delay in the arrival time of di ff erent pulse fre-quencies due to EP breaking leads to a shiftDM tot ( ˆ x , z ) → DM tot ( ˆ x , z ) + D grav ( ˆ x , z ) , (11)where D grav ( ˆ x , z ) is the EP-breaking time delay from eq. (8) inter-preted as a DM in direction ˆ x and up to redshift z : D grav ( ˆ x , z ) = ∆ γ K c (cid:16) ν − − ν − (cid:17) (cid:90) χ ( z )0 d χ (cid:48) a ( χ (cid:48) ) φ (cid:0) ˆ x χ (cid:48) , z ( χ (cid:48) ) (cid:1) . (12)As mentioned before, this contribution can be positive and negative,a problem which has not been addressed in previous studies. Notethat this identification is subject to the ν − law, thus providing apreferred frequency shape of the EP-breaking term. However, thenull hypothesis is ∆ γ =
0, as predicted by GR, and any additionalcontribution will immediately show up in the inferred DM budget. The non-gravitational contribution in eq. (11) is usually splitinto three parts:DM tot ( ˆ x , z ) = DM LSS ( ˆ x , z ) + DM MW ( ˆ x ) + DM host ( z ) . (13)( i ) For the contribution from the Milky Way, DM MW ( ˆ x ), models ofthe galactic electron distribution predict DM MW < ∼
50 pc cm − (Yaoet al. 2017). We will assume that this contribution can be modelledand subtracted from the signal. ( ii ) For the host galaxy contribution,DM host ( z ), similar values are expected inducing a random scatteron the total DM. ( iii ) The large-scale structure contribution can bewritten asDM LSS ( ˆ x , z ) = (cid:90) z n e ( x , z (cid:48) ) 1 + z (cid:48) H ( z (cid:48) ) d z (cid:48) . (14)Here H ( z ) is the Hubble expansion rate, and n e is the electron num-ber density. The latter is a functional of the matter density contrast δ m ( x , z ): n e ( x , z ) = ρ b ( x , z ) m p = ¯ ρ b ( z ) m p [1 + b e ( x , z ) δ m ( x , z )] , (15)with the baryon mass density ρ b ( x , z ), its mean value ¯ ρ b ( z ), the pro-ton mass m p and the electron clustering bias b e ( x , z ). We assumedthat the bias is linear, i.e. we consider only the lowest-order re-sponse of the electron density to the density of matter. This assump-tion should be valid on the scales we are considering. Rewritingeq. (15) yields:DM LSS ( ˆ x , z ) = A (cid:90) z d z (cid:48) + z (cid:48) E ( z (cid:48) ) F ( z (cid:48) )[1 + b e ( x , z (cid:48) ) δ m ( x , z (cid:48) )] , (16)with A = H Ω b0 χ H π Gm p , (17)where χ H is the Hubble radius today, E ( z ) = H ( z ) / H the dimen-sionless expansion rate, and F ( z ) the mass fraction of electrons inthe IGM, which itself can be expressed as follows: F ( z ) = f IGM ( z )[ Y H X e , H ( z ) + Y He X e , He ( z )] . (18)Here Y H = .
75 and Y He = .
25 are the mass fractions of hydrogenand helium, respectively, X e , H ( z ) and X e , He ( z ) are their ionizationfractions, and f IGM ( z ) is the fraction of electrons in the IGM. We as-sume X e , H = X e , He = f IGM ( z ) =
90% (80%) at z > ∼ . < ∼ . LSS ( ˆ x , z ) = DM LSS ( z ) + D LSS ( ˆ x , z ) . (19) D LSS ( ˆ x , z ) is the e ff ective DM induced by the fluctuations in theLSS and is given by the weighted line-of-sight integral over theelectron density perturbation in the second term of eq. (16). Notethat there is no background contribution for the gravitational part,eq. (12), which is exactly the problem in the ordinary measurementsof EP breaking with FRBs. The perturbations to the DM from theelectrons in the LSS and the Shapiro delay are therefore given by D ( ˆ x , z ) = D LSS ( ˆ x , z ) + D grav ( ˆ x , z ) . (20) MNRAS , 1–6 (2020)
Reischke, Hagstotz and Lilow ‘ . . . . . . . σ ∆ γ × − n tomo = 1 n tomo = 2 n tomo = 3 n tomo = 4 Figure 1.
Possible constraints on the EP-breaking parameter ∆ γ for a surveywith 10 FRB and α = . n tomo in the DMdistribution. Given a normalised source redshift distribution n ( z ), satisfy-ing (cid:82) d z n ( z ) =
1, and the associated distance distribution n ( χ ) = n ( z ) d z / d χ , eq. (20) can be averaged over redshift: D ( ˆ x ) = (cid:90) χ H d χ n ( χ ) D (cid:0) ˆ x , z ( χ ) (cid:1) . (21)By rearranging the integration limits we find D ( ˆ x ) = (cid:90) χ H d χ W D ( ˆ x , χ ) δ m (cid:0) ˆ x , z ( χ ) (cid:1) , (22)with the averaged weighting function W D ( χ ) = (cid:18) W LSS ( ˆ x , χ ) + W grav ( ˆ x , χ ) (cid:19) (cid:90) χ H χ d χ (cid:48) n ( χ (cid:48) ) . (23)Here, W LSS and W grav are defined via eq. (16) and eq. (12), respec-tively: W LSS ( ˆ x , χ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d z d χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A F (cid:0) z ( χ ) (cid:1) (cid:0) + z ( χ ) (cid:1) E (cid:0) z ( χ ) (cid:1) b e ( ˆ x , χ ) , (24) W grav ( ˆ x , χ ) = Ω m0 ∆ γ ∆ − K c χ H (cid:16) ν − − ν − (cid:17) , (25)where ∆ − is the inverse Laplacian relating the potential fluctua-tions φ to the matter density contrast δ m via the Poisson equation.The angular power spectrum of DM correlations for the source dis-tribution is then given by C DD ( (cid:96) ) = π (cid:90) d χ (cid:90) d χ (cid:90) k d k W D ( k , χ ) W D ( k , χ ) × (cid:112) P mm ( k , χ ) P mm ( k , χ ) j (cid:96) ( k χ ) j (cid:96) ( k χ ) , (26)where we express the weight function in Fourier space, so that theinverse Laplacian ∆ − = k − for a comoving wavenumber k . Theelectron bias b e is allowed to have a scale dependence. Note thateq. (26) depends on the two observed frequencies. Compared tothe sensitivity to ∆ γ this dependence is, however, very weak.The host galaxy acts as a stochastic source with an intrinsicwidth of σ . Due to the finite number of sources per solid angle,¯ n , this adds a white noise contribution to the observed spectrum: C DD ( (cid:96) ) → C DD ( (cid:96) ) + σ ¯ n . (27)As shown in Reischke et al. (2021), the shot noise contribution is small compared to cosmic variance on large scales even if only afew thousand FRBs are available. It should also be noted that anyredshift dependence of the host galaxy contribution to the DM hasbeen averaged out, weighted by the source redshift distribution. Weapproximate this distribution by the following form: n ( z ) ∝ z exp ( − α z ) , (28)where α determines the depth of the survey.In Reischke et al. (2021) the influence of the uncertainty on theDM in the absence of redshift information as well as its implicationfor the source redshift distribution and the corresponding angularpower spectra was discussed. The DM is translated into a redshiftassuming a fiducial cosmology without EP breaking. This can leadto dispersion space distortions (Masui & Sigurdson 2015) whichwe will, however, ignore here. However, the associated scatter ofthe redshift from the uncertainty of the DM is incorporated into theredshift distribution: n ( z ) = (cid:90) d z DM n ( z DM ) p ( z DM | z ) , (29)where p ( z DM | z ) is well approximated by a Gaussian distribution, asshown in (Jaroszynski 2019): p ( z DM | z ) ∼ N ( (cid:104) z DM (cid:105) ( z ) , σ z DM ( z ) ) (30)with corresponding mean (cid:104) z DM (cid:105) = (cid:104) DM host (cid:105) ( z ) + (cid:104) DM LSS (cid:105) ( z ) (31)and variance σ z DM = σ ( z ) + σ ( z ) . (32)The latter consists of the host contribution σ and the cosmolog-ical contribution σ .We want to stress that any ∆ γ (cid:44) Λ CDM scenario without EP breaking. In this sense we are assum-ing that we can convert a given DM into a corresponding redshifton the background level, while accounting for the scatter due tothe perturbations in the electron density and the host contribution(see Reischke et al. (2021) for more details). Rafiei-Ravandi et al.(2020) discussed how cross-correlations could be used to furtherimprove FRB distance measurements in the absence of redshift in-formation.Lastly it should be noted that everything could also be formu-lated in terms of the time delay directly, so that a time delay cor-relation function would be computed. This would be more closelyrelated to the e ff ects considered here. However, since FRB mea-surements are usually done in DM space, we stick to this notationhere. In this section we will discuss our results for a forecast of an EP nulltest. For this we assume a survey following the source distributioneq. (28) with α = .
5, which roughly corresponds to the distributionof currently observed FRBs (Petro ff et al. 2016). Note that we donot assume that the redshifts for the events are known, but only usethe probabilistic conversion from DM to z in eq. (29). Furthermore,we assume a total of 10 observed FRBs. Lastly, the electron bias isassumed to be b e = . z =
3, see (Reischkeet al. 2021) for more details.We consider an almost full sky survey, f sky = .
8, and decom-pose the DM map into spherical harmonics from which the angular
MNRAS000
MNRAS000 , 1–6 (2020) quivalence principle tests with FRBs signal-to-noise . . . . . σ ∆ γ × − Figure 2.
Constraints on the EP-breaking parameter ∆ γ as a function of thesignal-to-noise ratio of a measurement of the DM angular power spectrum. power spectrum can be estimated. The sensitivity with respect to ∆ γ is calculated using a Fisher forecast for a Gaussian likelihood of thespherical harmonic modes with covariance due to cosmic varianceand Poisson noise. The latter describes the intrinsic DM scattercaused by the host and the finite number of FRB sources. Addition-ally, the DM maps can be separated into sub-samples by consider-ing DM bins, as demonstrated in (Reischke et al. 2021). This meth-ods recovers some of the scale-dependent e ff ects that are smoothedout by the line-of-sight projection. A pulse width of 0.3 GHz is as-sumed (corresponding to 1.5 and 1.2 Ghz), which enters in eq. (25),and the host galaxy contribution is σ host ( z ) = − (1 + z ) − .In fig. 1 we show the resulting 1- σ limit on ∆ γ for the de-scribed survey as a function of the maximum multipole consid-ered and for di ff erent numbers of tomographic bins n tomo . The con-straints presented by Wei et al. (2015) are of the order of 10 − while Nusser (2016) found σ ∆ γ ∼ − for di ff erent FRBs. Theconstraints here are better by roughly two orders of magnitude.Since Nusser (2016) was interested in the rms-value and not thefull correlation, the sensitivity to ∆ γ is smaller than in the casestudied here, as the angular power spectrum contains terms scal-ing quadratically with ∆ γ . The highest impact of the Shapiro delaymeasurements can be seen at low multipoles, while the constrain-ing power settles very quickly at higher (cid:96) . This happens for two rea-sons: ( i ) the low multipoles are most a ff ected due to the k − factorin the Shapiro delay contribution coming from the Poisson equa-tion. Therefore, the contribution of the gravitational time delay willdominate the DM most strongly on the largest scales. ( ii ) Due tothe small number of FRBs, the shot noise starts dominating overthe signal at higher (cid:96) . Increasing the number of tomographic binsincreases the information content of the measurement to some de-gree, as it gets easier to pick up the scale-dependent imprint on theangular spectrum. However, the increased shot noise diminishes thegain when more than three bins are considered.To present the constraints independent of the specific surveysettings, we show in fig. 2 again the 1- σ limit on ∆ γ , but now asa function of the signal-to-noise ratio of the measurement of theangular power spectrum. It is clear from the figure that any sig-nificant detection of the angular correlation in the DM of FRBswill immediately put very stringent constraints on deviations fromthe EP, ∆ γ < − , with increasing signal-to-noise only yieldingmild improvements. The reason is that the integrated Shapiro delayover cosmological distances in combination with the short pulse durations yield C DD grav contributions to the angular spectrum that aremuch larger than any dispersion caused by electrons.For all forecasts made here we assumed all other parametersin the measurement to be fixed. Marginalizing over those, however,would not change the results dramatically since the e ff ect of theShapiro delay dominates the other contributions by many orders ofmagnitude if the EP is broken. In this work we re-investigated constraints on the breaking of theequivalence principle (EP), characterized by the post-Newtonianparameter di ff erence ∆ γ , with fast radio bursts by exploiting theShapiro time delay. Previous studies had focused on individualFRBs with redshift information by accounting for the Keplerian po-tentials of known objects along the line-of-sight. This was shownto be inconsistent by (Minazzoli et al. 2019). Here we proposed in-stead to use statistical properties of the dispersion measure (DM)of FRBs to constrain deviations from the EP. This measurementdoes not su ff er from divergences of the time delay and correctly ac-counts for the Shapiro delay induced by the LSS. In particular, welooked at the angular power spectrum of the DM of FRBs. Whilethe Shapiro delay along any given line-of-sight can be positive ornegative (causing problems with previous studies), in our approachthe imprint on the angular DM correlations can always be detectedas a change in the correlation structure. We performed a forecastfor a null test of EP violation by considering a fiducial Λ CDM cos-mology and placing upper limits on ∆ γ between observed frequen-cies. We would like to stress that the present parametrization is justa phenomenological one without a direct link to any specific EP-breaking theory. It is in this sense just an agnostic test of the degreeup to which the EP holds between two frequencies. We summarizeour principal results as follows:( i ) We calculated the contribution to the DM angular correla-tion function from the free electron distribution in the LSS and theinduced Shapiro delay by the LSS.( ii ) Using correlations of the DM measured from an FRB pop-ulation avoids the usual problems with Shapiro delay measure-ments by satisfying appropriate boundary conditions in a cosmo-logical setting.( iii ) Detecting the angular power spectrum of the FRB DMwith any significance at the predicted level will immediately placetight constraints on ∆ γ , which are a few orders of magnitudes betterthan the current limits.While we presented the theoretical predictions in DM space,one could alternatively formulate everything in time delay space,providing a closer connection to the Shapiro delay. Since the mea-surements of the angular correlation function will happen on verylarges scales, due to the relatively high shot noise contribution fromthe limited number of expected FRB detections, it will be neces-sary to calculate general relativistic projection e ff ects to the angu-lar power spectrum to make accurate theoretical predictions. Weintend to do this in a future paper. Due to the high sensitivity on ∆ γ this will, however, not change our main conclusion that EP canbe tested to unprecedented precision once the angular correlationof the FRB’s DM has been detected. MNRAS , 1–6 (2020)
Reischke, Hagstotz and Lilow
ACKNOWLEDGMENTS
RR is supported by the European Research Council (Grant No.770935). SH acknowledges support from the Vetenskapsrådet(Swedish Research Council) through contract No. 638-2013-8993and the Oskar Klein Centre for Cosmoparticle Physics. RL ac-knowledges support by a Technion fellowship.
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