Dark Neutrino interactions phase out the Hubble tension

New interactions of neutrinos can stop them from free streaming even after the weak interaction freezeout. This results in a phase shift in the cosmic microwave background (CMB) acoustic peaks which can alleviate the Hubble tension. We demonstrate with Planck CMB and WiggleZ galaxy survey data that this acoustic phase shift, and thus solution to the Hubble tension, can be achieved for neutrinos interacting with dark matter without significantly affecting other observables and without changing the number of relativistic degrees of freedom. We predict potentially observable modification of the CMB B-modes.

The values of the Hubble constant (H 0 ) inferred from cosmic microwave background (CMB) anisotropies (67.5 ± 0.6 km s −1 Mpc −1 [1,2]) and high redshift baryon acoustic oscillations (BAO) measurements (66.98±1.18 km s −1 Mpc −1 [3][4][5][6][7][8]) are significantly smaller than the measurements from observations of the nearby Universe using the distance ladder (74.03 ± 1.42 km s −1 Mpc −1 [9][10][11]). The gravitational lensing time delay measurements in multiply imaged quasar systems which are independent of the cosmic distance ladder also gives a higher value (72.5 +2.1 −2.3 km s −1 Mpc −1 [12,13]). This tension, calculated using Gaussian error bars, between the Planck CMB and local Hubble measurement stands at ∼ 4 σ [2,11], the exact number depending on the dataset used. Even though the recent independent recalibration of the cosmic distance ladder [14] which replaces the Cepheid variable stars based distances with Tip of the Red Giant Branch distances gives a value of H 0 slightly smaller than other local measurements, it has larger errorbar at present and also may have possible systematics [15]. Increasingly, this tension is being seen as a hint of physics beyond the ΛCDM cosmology , rather than a manifestation of possible systematics in the local distance ladder [45].
The spectacular success of the standard models of cosmology and particle physics in describing all cosmological and particle physics observables, however, makes the task of explaining the Hubble tension from new physics (NP) rather non-trivial. Particularly in this context, if the CMB data is to be reinterpreted with NP, the peaks and troughs of the power spectra must match data at least as well as the ΛCDM parametrization of the big bang cosmology. In fact, this condition alone neatly demonstrates the difficulty associated with introducing NP to solve the Hubble tension. The locations of acoustic peaks [46,47] in CMB data approximately correspond to the extrema of the cosine function characterizing the photon temperature transfer function, cos(kr * + φ), where k denotes the comoving wavenumber, r * is the comoving sound horizon at recombination, and φ is the phase shift with contribution (φ > 0) from free streaming neutrinos in ΛCDM cosmology [48]. The peak positions correspond to the wave numbers k peak which satisfy k peak r * = mπ − φ, where m ≥ 1 is an integer. The corresponding observed CMB peak multipoles ( peak ) are given by c s (z) is the speed of sound in the baryon-photon plasma, H(z) is the Hubble parameter, and D A is the comoving angular diameter distance to the redshift of recombination z * . Finding a solution to the Hubble tension requires keeping peak fixed while increasing H 0 . We see from Eq. (1) that we can modify the late time evolution of the Universe, i.e. modify H(z) for z < z * , in such a way that D A remains unchanged but H 0 ≡ H(0) is pushed higher, to reconcile CMB/BAO or acoustic H 0 with local H 0 [16, 18, 19, 22-25, 27, 28, 30, 32-34, 36-40]. Since in these solutions the early expansion history of the Universe ( H(z) for z > z * ) is unchanged r * remains unaltered. Therefore peak remain unchanged from the observed ΛCDM values. A second class of proposals rely on altering the cosmology before radiation domination, i.e. H(z) for z z * . These solutions change r * while at the same time keep r * /D A fixed [17,20,21,26,29,31,35,[41][42][43][44]. All of the solutions that have been proposed so far to alleviate the Hubble tension fall into the above two classes and, in particular, keep the acoustic scale at recombination θ * = r * /D A fixed even after accommodating a larger Hubble constant.
In this letter we find a new class of solutions where NP solves the Hubble tension by inducing changes in the phase shift φ and, therefore, are characterized by acoustic scales θ * different from that of the ΛCDM model. In order to understand the nature of NP that can accommodate a larger H 0 , let us consider a flat ΛCDM cosmology, with the Hubble parameter given by H(z) 2 where Ω i are the ratios of physical energy densities (ρ i ) to the critical energy density today and i = m, r for total non-relativistic matter and total radiation respectively. To separate out the effect of changing H 0 , lets keep the physical energy densities of matter and radiation, Ω m H 2 0 and is only important at low redshifts and becomes unimportant at high redshifts, when H(z) is much larger, and thus has negligible effect on r * . Therefore, we see from Eq. (1) that increasing H 0 (δ(H 2 0 ) > 0) decreases D A (δD A < 0). If δD A is to be compensated mostly from the shift in φ so that peak remains unaltered, we get from where we have explicitly used the notation δφ m to refer to the fact that the needed change in phase shift is different for different peaks. We have also used the fact that φ π in the second approximate equality. Therefore, if NP needs to accommodate a larger H 0 , it must induce a negative change in the phase shift that increases with m.
Incredibly, undoing the phase shift from free streaming neutrinos in the standard ΛCDM cosmology [48], produces almost exactly the required effect (see Fig. 1). Models where neutrinos carry beyond the standard model interactions, may allow neutrinos to scatter more and stop these from free-streaming, effectively generating a negative phase shift. Even though there exists a plethora of studies of cosmological impacts from nonstandard neutrinos interaction [26,29,[49][50][51][52][53][54][55][56][57][58][59][60][61][62], as well as studies of phase shift in the context of varying relativistic degrees of freedom (N eff ) on the phase shift [63][64][65], a detailed study of the impact of new neutrino interactions on acoustic phase shifts has not been performed yet.
In this work we present a simple proof-of-principle model, namely Dark Neutrino Interactions (DNI), where a component of dark matter interacts with neutrinos stopping them from free streaming. The DNI undo the phase-shift induced by the free streaming neutrinos in the standard model and thus push H 0 to higher values, and yet are safe from all cosmological and particle physics bounds. The necessary feature of this model is a two component dark matter. The total energy density of dark matter comes dominantly from a non-interacting standard cold dark matter (CDM) component. Only a small fraction,  Figure 1: CMB temperature (TT) power spectrum around first 4 acoustic peaks. The leftmost solid red line is the best fit Planck [66] temperature power spectrum with a best fit value of H 0 = 67.9 kms −1 Mpc −1 . Introducing DNI, keeping all other cosmological parameters fixed, moves all peaks to the right/higher with larger shift for higher peaks (rightmost solid blue curves). However DNI with higher H 0 brings the peaks back to the original positions (dashed blue). The amplitudes of DNI power spectra for each peak is adjusted so that the peak height is the same as the ΛCDM. Also shown as points with errorbars is the binned Planck power spectrum.
f , of the total dark matter energy density is contributed by the component that interacts with neutrinos or the neutrino interacting dark matter (NIDM). Note that having a small f allows us to evade the constraints typically obtained when all of the dark matter interacts with neutrinos [50,51,[54][55][56][57]. The primary ingredients for our model are therefore, (i) an interacting dark matter component, χ, (ii) a messenger, ψ, and (iii) an electroweak (EW) gauge invariant effective operator involving the Higgs scalar H and the lepton doublet l. After H acquires a nonzero vacuum expectation value (v) the effective operator gives marginal interactions among neutrinos, messengers, and dark matter.
where Λ is the scale of the effective operator and y is a dimensionless coupling constant. Note that we take ψ to be a flavor triplet and i, j in Eq. (3) are flavor indices. For a possible way to generate the interaction in Eq. (3) from a ultraviolet complete model using various symmetries see Ref. [62]. By construction, neutrinos are massless and all three flavors interact with equal strength.
In this work we focus on cases where the mediators and dark matter are nearly degenerate in mass. As shown in [62], this allows the scattering cross-section (σ χν ) between the dark matter and neutrinos to become independent of the neutrino temperature (T ν ). The temperature independence of DNI enables neutrinos to decouple late, undoing the phaseshift from free streaming neutrinos for all the modes entering horizon until recombination. We can write the "differential optical depth" for neutrinos in the DNI model aṡ where a is the scale factor, η is the conformal time, σ th = 6.65 × 10 −25 cm 2 is the Thomson cross-section and n χ , ρ χ , m χ denote the number density, the energy density, and the mass of χ respectively. Also, we have parametrized the interaction strength as The neutrino and the NIDM perturbation equations in DNI are coupled together [54] similar to the perturbations of the baryon-photon system and the initial conditions are also modified as the initial anisotropic stress is zero for tightly coupled neutrinos. We plot the ratio of interaction rate to Hubble rate,μ/(aH), in Fig. 2 for the current upper limits (f u = 0.034) for our model derived in this work. For comparison, we also show cases with neutrino self-interaction models [59,61] where crosssections vary as T 2 ν and T −2 ν . We see from Fig. 2 that with the current upper bounds (fixed N eff ) on neutrino interactions, we can stop the free streaming of neutrinos for all scales which enter horizon before recombination only in the temperature independent case.
We have implemented the DNI cosmology in publicly available code Cosmic Linear Anisotropy Solving System (CLASS) [67]. In DNI cosmology, the modes which enter horizon earlier (higher ) get a larger phase shift (w.r.t ΛCDM cosmology) compared to the modes which enter later as shown in Fig. 1 where we use f = 10 −3 , u = 34. This is because the relative contribution of neutrinos (∝ ρ ν /(ρ r + ρ m ), where ρ ν is the neutrino energy density) to the metric perturbations decreases with time as matter starts to dominate the energy density of the Universe. This is almost exactly the dependence that we need to solve the Hubble tension (Eq. (1)). We show this explicitly in Fig. 3 where we plot the (negative of) shift in peak positions for the CMB temperature and E-mode polarization angular power spectra as we change the Hubble constant in ΛCDM cosmology from the best fit value while keeping other parameters (Ω m H 2 0 etc) constant. For reference, we show the maximum effect we can get in the curve labelled "No ν-freestreaming" witḣ µ/(aH) ≫ 1. We see that the shift in peak for DNI cosmology, with the current upper bound in temperature independent interactions, is approximately of the same size (but in opposite direction) as ΛCDM cosmology with H 0 = 70km/s/Mpc. The scalings in are also similar in both the cases . Therefore, we expect that the Hubble tension should reduce considerably in a DNI cosmology. We verify this in the DNI curves with H 0 = 70km/s/Mpc, in which the peak shifts are negligible compared to the best fit Planck ΛCDM cosmology.
We perform a Markov-Chain Monte Carlo (MCMC) analysis of the DNI model using publicly available code Monte-Python [68]. We use the following cosmological data sets: Planck CMB 2015 Low-TEB, High TT EE TE -Plik lite and CMB lensing T+P [66] (named 'P15') and full shape of Galaxy power spectrum measured by WiggleZ Dark Energy Survey [69]. The WiggleZ power spectrum goes upto k = 0.5 h Mpc −1 . We have used different k-cutoff of the full dataset for three separate analyses and label them W1, W2, W3 for cutoff k max = 0.12h, 0.2h, 0.3h Mpc −1 respectively, where h ≡ H 0 /(100 kms −1 Mpc −1 ). We used CLASS Halofit module [70] to incorporate non-linear modifications in the power spectrum.
Note on BAO data: It will be incorrect to use just the BAO scale (or θ * ) extracted from the power spectrum [e.g. 8] assuming ΛCDM cosmology, available as BAO likelihood modules in public MCMC codes, to constrain any new physics which modifies the phase shift φ of the acoustic oscillations and allows θ * to vary from the ΛCDM value. This is the case for us and also for any model with non-standard N eff , since any new free streaming relativistic species contributes to φ in a scale dependent manner.
The results of our MCMC analysis, with two extra DNI parameters f and u are shown in the left panel of Fig. 4, where we show constraints in the (H 0 − f u) plane while marginalizing over ΛCDM parameters. The local measurement from [11] of H 0 = 74.03 ± 1.42 km s −1 Mpc −1 is shown in gray horizontal bands. There is a clear degeneracy between the neutrino stopping power (∝ f u) and H 0 which reduces the Hubble tension. We see from the MCMC samples plotted in Fig. 4 (centre) that stronger neutrino interaction favours higher H 0 .
The 1-D probability distribution functions (PDF) shown in inset of 4 (left) are highly non-Gaussian. To quantify the tension between non-Gaussian PDFs, we define a quantity d = (H 1 − H 2 )/ σ 1 (t) 2 + σ 2 (t) 2 , where H 1 , H 2 are two H 0 measurements and σ 1 (t), σ 2 (t) are the corresponding 't-σ' upper or lower limits. For a Gaussian PDF σ(t) = tσ G , where σ G is the Gaussian 1-σ error. We use Gaussian errorbar for the local H 0 measurement and plot the quantity d in Fig. 4 (right). The tension is then given by the value of t where d = 1. Our definition is equivalent to the usual definition of tension in the Gaussian case. We see that for ΛCDM the tension is at 3.8σ which reduces to 3σ in DNI cosmology. There is a small second peak for the 'P15 + W1' dataset in the inset of Fig. 4 (left) which shows up as the red disconnected patch within 3σ contour in the 2D plot. This results in a big jump in d and reduces the tension to 2.1σ.
In Table 1 we present results of a MCMC analysis of DNI cosmology for fixed f = 10 −3 where we also include the local measurement of H 0 (SH0ES collaboration [11]). With respect to ΛCDM, χ 2 reduces by 9 in DNI with one extra parameter u. The bestfit value of the Hubble constant turns out to be H 0 = 70.2. Note that, as argued before, this increase in H 0 is associated with a decrease in D A which in turn gets compensated mostly from a change in φ. Therefore, the bestfit for DNI cosmology is characterized by a θ * which is ∼ 15σ away from that of ΛCDM. There is however a small change in r * which roughly compensates ∼ 20% change in D A . Interestingly, DNI cosmology is a slightly better fit to the 'P15 + W1' datasets than the ΛCDM cosmology. Note that we have used W1 cutoff to avoid the non-linear scales.
We see in Table. 1 that the bestfit f u ≈ 2 × 10 −2 requires u ≈ 20. For m χ 1 MeV we find the scale of the effective operator to be Λ 2.5 TeV from Eq. (5)    a Λ we do not expect any significant constraint from particle physics. The gravity of new neutrino interactions modifies the B-mode CMB power spectrum [62]. We compare in Fig. 5 the modification of B-modes for tensor to scalar ratio r = 0.06 [1,72,73] with the sensitivity of the proposed experiment Polarized Radiation Imaging and Spectroscopy Mission (PRISM) [71]. This effect, in principle, can be detected if r is close to the current upper limit [1,72,73] with a future PRISM like experiment [71,[74][75][76][77].
In this work we have proposed a qualitatively new framework that ameliorates the Hubble tension by using the phase shift in the acoustic oscillations of the primordial plasma. Amazingly, this framework undoes the neutrino induced phase-shift of ΛCDM, gives the correct shift in the acoustic peaks of CMB and BAO and pushes the acoustic H 0 higher reconciling it with the local H 0 . We therefore might have detected new interactions of neutrinos in the Hubble tension.