^$^$institutetext: Citi Bank^b^binstitutetext: Texas Center for Cosmology and Astroparticle Physics, Weinberg Institute, Department of Physics, The Unversity of Texas at Austin, Austin, TX 78712, USA^c^cinstitutetext: Department of Physics, University of Notre Dame, IN 46556, USA^d^dinstitutetext: PITT PACC, Department of Physics and Astronomy,
University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15260, USA

A cosmological case study of a tower of warm dark matter states: $N$ naturalness

Saurabh Bansal, b,c Subhajit Ghosh d Matthew Low c and Yuhsin Tsai [email protected] [email protected] [email protected] [email protected]

Abstract

In this work, we studied the cosmological effects of a tower of warm dark matter (WDM) states on the cosmic microwave background (CMB) and large-scale structure (LSS). For concreteness, we considered the $N$ naturalness model, which is a proposed mechanism to solve the Electroweak Hierarchy Problem. In this framework, the particles of the Standard Model are copied $N$ times where the Higgs mass-squared value is the only parameter that changes between sectors. The other sectors are similar to our own, except their particles are proportionally heavier and cooler compared to the Standard Model sector. Since each sector is very weakly coupled to other sectors, direct observations of the new particles are not expected. The addition of $N$ photons and $N$ neutrinos, however, can be detected through the CMB and in LSS data. The $N$ -neutrinos form a tower of states with increasing mass and decreasing temperature compared to SM neutrinos. This tower causes a more gradual suppression of the matter power spectrum across different comoving wavenumbers than a single WDM state would. We quantitatively explored these effects in the $N$ naturalness model and presented the parameter space allowed by the Planck 2018, weak lensing, and Lyman- $\alpha$ datasets. Depending on the reheaton mass, Planck+BAO data can require tuning of the model to as much as $\approx 3\%$ ( $10\%$ ) for Majorana (Dirac) neutrinos. The $N$ -neutrinos are crucial in constraining the model, particularly through their suppression of the power spectrum at small scales. We also highlight the need for a faster Boltzmann solver to study the impact of the neutrino tower on smaller scales.

1 Introduction

The early Universe serves as a unique laboratory for exploring new physics across a broad range of energy scales. At extremely high energies, new physics occurring during inflation can modify curvature perturbations and imprint signatures on the cosmic microwave background (CMB) and large-scale structure (LSS). We have increasingly precise measurements of both of these phenomena from past and current surveys. Near the MeV scale, new physics can be constrained through measurements of the number of additional relativistic degrees during Big Bang Nucleosynthesis (BBN). Below the KeV scale, new physics, in the form of self-interactions among dark sector particles or neutrino interactions can impact the CMB and structure formation. Finally, below $\sim 100$ eV precise measurements of the CMB temperature spectrum can probe tiny energy injections or leakage from the photon-baryon plasma. ML: I think this paragraph needs to be cleaned up since it’s the opening paragraph.

Notably, none of these energy scales directly address the electroweak scale, which harbors the enduring mystery of the Higgs hierarchy problem. This problem has motivated extensive efforts to explain the absence of new physics signals at the Large Hadron Collider (LHC), where one would expect the presence of new physics to mitigate the quadratic divergence arising from quantum corrections to the Higgs mass. Despite this fact, cosmological observations have been shown to constrain – and in certain cases even measure – some solutions to the Higgs hierarchy problem, leading to a strong complementarity with collider searches.

For instance, consider the Mirror Twin Higgs (MTH) model Chacko:2005pe, which addresses the Higgs hierarchy problem up to the 10 TeV scale. In this model there is a concretely predicted spectrum of mirror sector particles with all cosmological properties fixed by three parameters: the temperature of the mirror sector, the baryon abundance in the mirror sector, and the tuning required for the MTH model to solve the Higgs hierarchy problem. The model predicts several cosmological signatures, including dark acoustic oscillations between mirror baryons and photons and the presence of both free-streaming and interacting radiation in the form of mirror neutrinos and mirror photons Bansal:2021dfh; Bansal:2022qbi. These new physics ingredients coming from the mirror symmetry that addresses the hierarchy problem leave distinct signalsimprints on the CMB and matter power spectra as the SM baryon and photon. As a result, cosmological data from the Universe below a temperature of $\sim 100$ eV have imposed useful constraints on the MTH solution to a 100 GeV scale puzzle. The cosmological study of mirror-sector models is not only an interesting case study; it has real applications to well-motivated particle physics scenarios. ML: I think the connection between cosmology scales and Higgs scales is not clear from this paragraph [YT: I added a few words to emphasize that it’s the mirror symmetry that produces SM-like particles and signals similar to the SM sector. Can this help to explain why cosmo data is relevant to the MTH model that addresses the Higgs scale problem?]

In this work, we explore the cosmology of another solution to the Higgs Hierarchy problem, focusing on the cosmological signatures from a dark sector containing dark radiation and a spectrum of warm dark matter (WDM) states. The $N$ naturalness model Arkani-Hamed:2016rle; Banerjee:2016suz; Choi:2018gho is proposed to explain the separation between the electroweak scale and a much higher energy scale, where new physics must emerge to address problematic quantum corrections to the Higgs mass. As we review later, this model includes $N$ copies of Standard Model (SM) sectors, differing only in the parameter of the Higgs mass. Assuming no baryon asymmetry in the other SM-like sectors, the minimal additions to late Universe cosmology are dark radiation from $N$ -sector photons and massless leptons, plus a tower of WDM corresponding to the three neutrinos from all $N$ SM sectors. Thus, the details of $N$ naturalness cosmology are determined by two parameters: the mass of a reheaton that reheats the Universe, and the tuning that aligns the sector with the minimum Higgs mass to our $\sim 100$ GeV scale SM. Beyond addressing the hierarchy problem, the QCD phase transition in some $N$ -sectors could be first-order, producing gravitational waves detectable in future experiments Batell:2023wdb. Earlier studies estimated the bound on the model simply based on the additional relativistic degrees of freedom $\Delta N_{\rm eff}$ from the massless radiation Banerjee:2016suz; Choi:2018gho. However, the thermal history of the $N$ -sector neutrinos can be quite complex and has significant impact on the cosmological bounds Arkani-Hamed:2016rle; Choi:2018gho. As we will show, the presence of $N$ WDM with different but predicted masses, temperature, and abundance alter the CMB and LSS signals, allowing us to set much stronger constraints on this solution to the electroweak scale hierarchy problem.

While we use a solution to the Higgs hierarchy problem to motivate a cosmology with non-photon radiation and multiple WDM states, a similar setup exist even for SM neutrinos, which can have one massless and two massive neutrinos with a normal hierarchy mass spectrum. Studies have shown that the difference in the distribution of masses for normal and inverted hierarchies has observable effects on matter power spectrum Agarwal_2010. The presence of additional sterile neutrinos with thermal or non-thermal distribution makes the resemblance with multiple-WDM setup more prominent. For dark sector models, this is also a straightforward extension of single WDM scenarios and has been considered in various new physics contexts.[SG: Find some more references][SG: The meaning of this line is not clear]. Decays within multiple dark sector WDM states can induce non-trivial changes in the dark matter phase-space distribution which has rich effects on matter power spectrum Dienes:2020bmn; Dienes:2021cxp; Dienes:2021itb.[SG: Find more reference from Kieth et. al.]

However, the cosmological signatures of multiple-WDM models, as opposed to single WDM scenarios, are much less explored. Specifically, the presence of a tower of WDM states induces salient features, such as novel scale-dependent suppression of matter power spectrum and scale-dependent $\Delta N_{\rm eff}$ , which has not been systematically studied in the literature. [YT: check if Keith already studied these]. Here, we use the $N$ naturalness model as a concrete example to study the cosmological effects of the tower of neutrino states on CMB and LSS. As we will show, compared to a single WDM model that produces the same $\Delta N_{\rm eff}$ and the same dark matter abundance today, a tower of WDM states results in power spectrum suppression at comparatively smaller length scales. This leads to a distinct shape in the matter power spectrum compared to the single WDM scenario.

Though our study focuses on a specific model with a defined relation between sector number and neutrino mass and temperature, the cosmological effects we discuss are generally similar in models with multiple WDM states that become non-relativistic at different times. The strong bound on $\Delta N_{\rm eff}$ and measurement of DM abundance from cosmology generally strongly constrain the mass and temperature profile/scaling of multiple WDM states, which are similar to that of the $N$ naturalness. One common challenge in studying models with many WDM states is the significant computational power required to solve the phase space integral for each WDM particle. The higher the comoving wavenumber ( $k$ ) of a perturbation we want to calculate, the more WDM species we need to include to reliably predict the signal. Therefore, we mainly focus on observables with $k\lesssim 0.3$ Mpc^-1 and appropriately truncate the neutrino tower based on their mass and kinetic energy evolution.[SG: the last part of the sentence?] Nonetheless, we also explore the sensitivity of small scale Lyman- $\alpha$ data in constraining the $N$ -neutrino sectors under certain relaxed parameter assumptions.

The paper is organized as follows. In Sec. 2 we review the $N$ naturalness model and its parameters. We describe the implementation of the model in CLASS in Sec. 3, and discuss the behavior of CMB and matter power spectra from the presence of $N$ -photon and $N$ -neutrino in Sec. 2 and Sec. 4. In Sec. 5, we study the difference in matter power spectrum suppression between one and $N$ WDM species. We conduct the Markov-Chain Monte Carlo (MCMC) analysis in Sec. 6 and show that the inclusion of $N$ -neutrino dynamics strengthens the $N$ naturalness constraint significantly. In Sec. 7, we estimate the sensitivity of Lyman- $\alpha$ constraint to the model. We conclude in Sec. 8.

Citations to add: NN gravitational waves Archer-Smith:2019gzq, other constraints Baumgart:2021ptt.[SG: Add these to earlier citations?]

2 The Model

In this section we describe the general features of models of $N$ naturalness and the specific model we use in our study.

Sectors

In the $N$ naturalness framework there are $N$ sectors of particles. Each sector, labelled by $i$ , is assumed to be the same except for its $m_{H,i}^{2}$ parameter. If these parameters are allowed to vary with uniform spacing, their values are

m_{H,i}^{2}=-\frac{\Lambda_{H}^{2}}{N}(2i+r),\quad\quad\quad-\frac{N}{2}\leq i\leq\frac{N}{2},

(1)

where $\Lambda_{H}$ is the cut-off. The Standard Model is identified to be the $i=0$ sector with $m_{H}^{2}=m_{H,0}^{2}=-(88~{}{\rm GeV})^{2}$ . The parameter $r$ controls how close the sector with the smallest negative $m_{H}^{2}$ value is to zero and is a good proxy for the amount of fine-tuning in the model. When $r\ll 1$ the theory is considered tuned, meaning that it does not address the hierarchy problem well, because the cut-off is $1/\sqrt{r}$ larger than the naive expectation. Equivalently, when $r\ll 1$ , $m_{H}^{2}$ is closer to zero than naively expected.

The inclusion of $N$ sectors has two immediate consequences. Firstly, it renormalizes the scale where gravity becomes strong to $M_{\rm Pl}/\sqrt{N}$ . Secondly, due to Eq. (1), the lightest negative $m_{H}^{2}$ parameter is approximately $\Lambda_{H}^{2}/N$ , which is substantially lower than the squared cut-off $\Lambda_{H}^{2}$ when $N$ is large.

Choosing $N\sim 10^{16}$ would raise $\Lambda_{H}$ to $\sim 10^{11}~{}{\rm GeV}$ while simultaneously lowering the gravitational scale to the same value, thus solving the full hierarchy problem. Smaller values of $N$ would leave a gap between $\Lambda_{H}$ and the gravitational scale and would require another fine-tuning solution, like supersymmetry, between those scales. The value of $N\sim 10^{4}$ is a natural benchmark because it raises $\Lambda_{H}$ enough to solve the little hierarchy problem while pushing the gravitational scale down to coincide with the scale of grand unification Arkani-Hamed:2016rle.

Sectors with $m_{H}^{2}<0$ experience electroweak symmetry breaking in the same way as the SM and we call these SM-like sectors. The more negative the value of $m_{H,i}^{2}$ in a SM-like sector $i$ , the larger the vacuum expectation value (VEV) $v_{i}$ of that sector is compared to that of the SM $v_{\rm SM}$ . The particle spectrum, with the possible exception of neutrinos, in an SM-like sector $i$ is the same as the SM but with masses that are larger by a factor of $v_{i}/v_{\rm SM}$ . Dirac neutrinos follow this scaling, but Majorana neutrinos are an exception and scale as $(v_{i}/v_{\rm SM})^{2}$ .

Sectors with $m_{H}^{2}>0$ , on the other hand, do not experience electroweak symmetry breaking via the Higgs potential. Instead, in these sectors, the chiral condensate of the corresponding QCD sector breaks electroweak symmetry at a much lower scale of $f_{\pi}\sim 100~{}{\rm MeV}$ .¹¹1Due to the differences in quark masses, the running of $\alpha_{s}$ in each sector is slightly different resulting in slightly different values of $f_{\pi}$ . The variation, however, is very mild so we refer to a single value of $f_{\pi}$ for simplicity. The massive vector bosons have masses of $\sim gf_{\pi}\sim 100~{}{\rm MeV}$ , rather than $\sim 100~{}{\rm GeV}$ . A Higgs VEV is induced through the couplings of the Higgs to the chiral condensate which leads to very light fermion masses of $\sim y_{t}y_{f}\Lambda_{\rm QCD}^{3}/m_{H,i}^{2}$ , where $y_{f}$ is the Yukawa coupling of a fermion $f$ . We call these exotic sectors since their dynamics and cosmology is different from that from that of the SM.

Reheating

In order to have a viable cosmological history, the SM sector must be dominantly populated. This is accomplished through a particle called the reheaton which is assumed to dominate the energy density of the universe at some early time. The reheaton then decays into all available channels, dominantly populating the SM, but also leaving potentially measurable amounts of energy density in the other sectors.

In this work, we consider the scalar reheaton model where a new real scalar $\phi$ is added

\mathcal{L}_{\phi}\supset\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{m_{\phi}^{2}}{2}\phi^{2}-a_{\phi}\phi\sum_{i}|H_{i}|^{2}.

(2)

The mass of the reheaton is $m_{\phi}$ and $a_{\phi}$ is a trilinear term that couples $\phi$ and two Higgs particles from each sector with the same magnitude for all sectors.²²2To maintain a well-behaved theory in the large $N$ limit $a_{\phi}$ should have an arbitrary sign for each sector with a magnitude that scales as $|a_{\phi}|\lesssim\Lambda_{H}/N$ .

The trilinear term results in mixing between each Higgs that undergoes electroweak symmetry breaking and $\phi$ . The mixing between the Higgs of the SM sector and $\phi$ leads to an upper bound on $a_{\phi}$ from Higgs mixing constraints of roughly $|a_{\phi}|\lesssim 1~{}{\rm MeV}$ Arkani-Hamed:2016rle. Only ratios of couplings enter into cosmological observables making these observations nearly insensitive to the value of $a_{\phi}$ .

Decays into SM-like sectors therefore proceed via mixing with the Higgs of that sector. Partial widths scale as $\Gamma_{i}\sim 1/m_{h,i}^{2}$ and the number of kinematically open decay channels decreases as $i$ increases because the mass scale of SM-like sectors gets higher and higher. The energy density $\rho_{i}$ in a sector $i$ will scale as $\rho_{i}/\rho_{\rm SM}\approx\Gamma_{i}/\Gamma_{\rm SM}$ which means that only a handful of sectors have a relative energy density larger than the per mille level. The total energy density, however, is approximately logarithmically sensitive to $i$ until $i$ is large enough where the number of open decay channels drops substantially.

At the scale of the reheaton mass, electroweak symmetry in the exotic sectors is unbroken so there is no mixing between the reheaton and each Higgs of the exotic sectors. Decays into these sectors, instead, proceeds dominantly through a loop of Higgses into a pair of vectors Arkani-Hamed:2016rle. When $m_{H}<m_{\phi}<2m_{H}$ the three-body decay of $\phi\to Ht_{R}t_{L}$ is possible, but is only numerically relevant when $m_{\phi}$ is very close to $2m_{H}$ Batell:2023wdb. The loop decay leads to a partial width that scales as $\Gamma_{i}\sim 1/m_{H,i}^{4}$ . The energy density in the exotic sectors is negligible in most of the parameter space. If the range of $r$ is extended to larger than $1$ then there are small regions where the $i=-1$ can have a measurable energy density Batell:2023wdb.

Parameter Space

The theory is characterized by the number of sectors $N$ , the reheaton mass $m_{\phi}$ , the spacing parameter $r$ , the trilinear coupling $a_{\phi}$ , and whether the neutrinos are Majorana or Dirac. The properties of $N$ naturalness cosmology, however, are determined by a subset of these parameters: the continuous parameters $m_{\phi}$ and $r$ , the binary choice of Majorana or Dirac neutrinos, and a discrete set of values of $N_{\nu}\leq N/2$ .

Since cosmological observables depend on the ratio of reheaton partial widths they are almost entirely insensitive to the value of $a_{\phi}$ . For concreteness we set $a_{\phi}=1~{}{\rm MeV}$ . The ratio of reheaton partial widths does depend on both $m_{\phi}$ and $r$ . We consider values of $m_{\phi}$ between 5 GeV and 300 GeV. Above 300 GeV the reheaton can start to decay on-shell to pairs of SM-like Higgses in the lightest sectors which disrupts the $N$ -Naturalness mechanism. For $r$ , we consider values between 0 and 1.³³3In Ref. Batell:2023wdb it was shown that the range of $r$ can be consistently extended up to 2.

Of the new particles added in $N$ naturalness, the cosmologically relevant ones are the photons from the additional sectors and the neutrinos from the additional sectors. We refer to collection of these from all new sectors, respectively, as the $N$ -photons and $N$ -neutrinos. For the $N$ -photons we fix the number of sectors to $N=10^{4}$ which includes $N/2$ SM-like sectors and $N/2$ exotic sectors. This value addresses the little hierarchy problem and is consistent with overclosure bounds of from the total density of cold dark matter (CDM) Arkani-Hamed:2016rle. Results are very insensitive to the value of $N$ as sectors with large $i$ have very suppressed contributions to $\Delta N_{\rm eff}$ .

Regarding the $N$ -neutrinos, we only include the neutrinos from $N_{\nu}$ SM-like sectors, where $N_{\nu}\leq N/2$ . Computationally it is intractable to solve the Boltzmann equations for such a large number of neutrino species so we aim to use the smallest value of $N_{\nu}$ that leads to accurate results. We are able to perform calculations for values of $N_{\nu}\lesssim 50$ , but we study in detail the estimated impact of truncating the heavy neutrino tower. To our advantage, we will show that neutrinos with sector $i\gtrsim 100$ can be safely treated as CDM for even the smallest scale cosmological measurements. In most cases we use $N_{\nu}=20$ as the default benchmark, with the exception of our Lyman- $\alpha$ analysis which requires $N_{\nu}=50$ .

Finally, in all cases we consider Majorana and Dirac neutrinos separately. Their behavior noticeably differs because the mass scale of the Majorana neutrinos grows much faster. ML: Add another sentence about the general differences.

$N$ -photon Signals

The impact of the $N$ -photons on cosmological observables is primarily encapsulated by the contribution to the number of relativistic degrees of freedom, $\Delta N_{\rm eff}$ . The photons from the SM-like sectors are free-streaming and their contribution can be approximated as

\Delta N_{{\rm eff},i>0}=\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}\frac{1}{\Gamma_{\rm SM}}\sum_{i>0}\Gamma_{i},

(3)

where $\Gamma_{\rm SM}$ is the partial width of the reheaton decaying into all kinematically accessible states of the SM and $\Gamma_{i}$ is the partial width of the reheaton decaying into all kinematically accessible states of sector $i$ . Eq. (3) neglects the mild dependence that arises from changes in the degrees of freedom $g_{*,i}$ in a sector $i$ as a function of temperature. This was calculated in Ref. Batell:2023wdb and found to only introduce $\mathcal{O}(20\%)$ corrections to the $\Delta N_{\rm eff}$ calculation over the model parameter space we study.

The photons from the exotic sectors may be interacting at the time of the CMB because the corresponding electrons are much lighter and stay in thermal equilibrium with their respective photons until much later times. Additionally, the exotic sectors likely undergo first order QCD phase transitions⁴⁴4Earlier studies, e.g. Refs. PhysRevD.29.338; Butti:2003nu; Iwasaki:1995ij; Karsch:2003jg suggested that QCD with vanishing quark masses would undergo a first order phase transition, but some modern lattice results disfavor this claim Cuteri:2021ikv. which result in an increase in energy density from the phase transition Batell:2023wdb. For a phase transition of strength $\alpha$ , which is defined as the ratio between the change of QCD vacuum energy change and the total energy density in each sector, the energy density increases roughly by a factor of $1+\alpha$ , where typically values of $\alpha$ may range from $0.05-10$ Helmboldt:2019pan; Bigazzi:2020avc; Reichert:2021cvs. In this work we safely neglect these effects as our parameter space does not contain regions where the energy density from exotic sectors is non-negligible.⁵⁵5To validate our statement that the exotic sector energy density is negligible we can compare $\Delta N_{{\rm eff},i>0}$ from Eq. (3) with the same summation over the exotic sectors $\Delta N_{{\rm eff},i<0}$ . When the ratio of $(\Delta N_{{\rm eff},i<0})/(\Delta N_{{\rm eff},i>0})>1$ the value of $\Delta N_{\rm eff}>0.58$ . For ratios of $0.1,0.01,0.001$ the corresponding limits on $\Delta N_{\rm eff}$ are $0.22,0.13,0.09$ , respectively. Therefore any parameter space where the relative energy density in the exotic sectors is large also has a large value of $\Delta N_{\rm eff}$ .

In our study, we use $\Delta N_{\rm eff}^{\gamma}=\Delta N_{{\rm eff},i>0}$ . The values of $\Delta N_{\rm eff}^{\gamma}$ in the $(m_{\phi},r)$ parameter space are shown in Fig. 1. This is the only bound considered in Ref. Arkani-Hamed:2016rle and serves as a useful benchmark to compare the impact that constraints from other cosmological observables, especially neutrinos, have on the $N$ naturalness parameter space. The current bound from the CMB is $\Delta N_{\rm eff}\lesssim 0.5$ which includes data from the SH0ES collaboration Planck:2018vyg; Blinov:2020hmc; Riess:2021jrx. The naive $\Delta N_{\rm eff}^{\gamma}$ constraint, for most $m_{\phi}$ values, requires $r\lesssim 0.2-0.4$ , corresponding to a mild $20-40\%$ tuning in the model.

The bounds become significantly weaker in three $m_{\phi}$ regions either to due to a larger value of $\Gamma_{\rm SM}$ or a smaller value of $\sum_{i}\Gamma_{i}$ , as seen in Eq. (3):. These regions are the following. (1) Small $m_{\phi}\lesssim 20$ GeV. Here, $m_{\phi}$ decays predominantly into fermions. Decays of the reheaton to two SM bottom quarks is open, but depending on $r$ , many of the decays to $i>0$ bottom quarks become kinematically inaccessible. (2) $m_{\phi}$ close to 125 GeV which is the mass of the SM Higgs. In this case, the reheaton has substantial mixing with the SM Higgs, resulting in dominant decay into the SM sector. (3) $m_{\phi}$ above $\approx 160$ GeV, which is twice the mass of the SM $W$ boson. Here, the reheaton can decay into two SM $W$ bosons with a large branching ratio.

Refer to caption — Figure 1: The number of effective relativistic degrees of freedom, only from photons, $\Delta N_{\rm eff}^{\gamma}$ , in the plane of $r$ and $m_{\phi}$ .

3 Implementation in CLASS

In order to calculate the detailed cosmological signals of the $N$ naturalness model, we have modified the Boltzmann solver code Cosmic Linear Anisotropy Solving System (CLASS) Diego_Blas_2011. In this section, we comment on the subtleties and the approximations of our CLASS implementation.

We have assumed a negligible baryon symmetry in all SM-like and exotic sectors. This is a motivated choice both in light of overclosure bounds and in light of baryogenesis considerations Arkani-Hamed:2016rle. Relaxing the constraint will imply that part of the total CDM today is constituted by the baryons and leptons of the SM-like and exotic sectors. The cosmological effects of the additional sectors are, therefore, manifested through photons and neutrinos in the additional sectors. Roughly speaking, the additional photons contribute as dark radiation and the additional neutrinos contribute as warm dark matter when relatively light and as CDM when relatively heavy. These contributions are calculated for each point in the $N$ naturalness parameter space.

The additional photons impact cosmological evolution through their contribution to $N_{\rm eff}$ which we denote as $\Delta N_{\rm eff}^{\gamma}$ . We calculate $\Delta N_{\rm eff}^{\gamma}$ analytically outside of CLASS which includes a summation over all SM-like sectors.

The impact of an additional species of neutrino depends on the mass of the species. When it is sufficiently heavy, it acts as CDM and contributes to the CDM abundance. At lighter masses, however, each neutrino species acts as warm dark matter and its impact should be included by solving the momentum-dependent Boltzmann equations. Therefore, we should be able to include a subset of the additional neutrino species in the Boltzmann equations and adjust the energy density of CDM without sacrificing any accuracy. As mentioned in the previous section, $N_{\nu}$ is the number of sectors whose neutrinos are included in the full Boltzmann equations. Solving the Boltzmann equations for the additional neutrinos requires the temperature and masses of the neutrinos from each sector. For a given point in parameter space, the neutrino temperatures and masses are input into CLASS through a precomputed table schematically represented in Table 1.

Solving the Boltzmann equations of momentum-dependent massive neutrinos is highly computationally intensive Kamionkowski:2021njk. For each sector of neutrinos there are three neutrino species requiring $3N_{\nu}$ sets of Boltzmann equations. For $N_{\nu}\approx N/2\gg 1$ running the code becomes intractable. We make the following approximations. Firstly, we assume that the three species of neutrinos in sector are degenerate within that sector. This is achieved in CLASS by setting the flag deg_ncdm = 3. The SM neutrinos are also taken to be degenerate with a sum of masses of $m_{\nu,{\rm SM}}=0.12~{}{\rm eV}$ , which is the well-established $2\sigma$ upper bound from Planck and BAO datasets Planck:2018vyg. Although degenerate-mass neutrinos are incompatible with neutrino oscillations, the primary cosmological signatures of neutrinos are the sum of neutrino masses and their contribution to $\Delta N_{\rm eff}$ . In particular, the case of three neutrinos per sector $i$ with masses $\{m^{(1)}_{\nu,i},m^{(2)}_{\nu,i},m^{(3)}_{\nu,i}\}$ and the case of a single triply-degenerate neutrino per sector $i$ with a mass of $m_{\nu,i}=(m^{(1)}_{\nu,i}+m^{(2)}_{\nu,i}+m^{(3)}_{\nu,i})/3$ are nearly indistinguishable when the total neutrino mass for a sector is $\gtrsim 0.2$ eV which is nearly already the case for our choice of SM neutrino masses Lesgourgues:2006nd.

Secondly, we set precision parameters in CLASS as l_max_ncdm = 5 (the default is 17) and tol_ncdm_synchronous = 0.01 (the default is 0.001). We checked this lower precision setting induces $0.6\%$ changes in the CLASS anisotropy spectrum which are below the precision of the Planck data at the corresponding multipole range. Finally, we do not use $N_{\nu}=N/2$ with $N=10^{4}$ , but use a smaller value. In practice, we use values $N_{\nu}=20$ for CMB and $N_{\nu}=50$ for LSS (Lyman- $\alpha$ ) for an accurate approximation of the neutrino effects. These approximations are studied in detail in Sec. 4.

We summarize the approach below. As discussed in Sec. 2, a point in the $N$ naturalness parameter space is parameterized by the values of $r$ and $m_{\phi}$ . The value of $r$ determines the spacing between the additional sectors and the value of $m_{\phi}$ impacts the temperature of each sector via the branching ratios of the reheaton. For each parameter point, we compute $\Delta N_{\rm eff}^{\gamma}$ from the $N$ -photons from all SM-like sectors and we compute $T_{\nu,i}$ and $m_{\nu,i}$ for each SM-like sector. These values are stored in a table which is read into CLASS. Table 1 shows the schematic form of this table. When augmented with this table, our modified version of CLASS acts as a two-parameter extension of $\Lambda$ CDM.

$r$	$m_{\phi}$	$\Delta N_{{\rm eff},\gamma}$	$i=1$				$i=2$				$\cdots$
$r$	$m_{\phi}$	$\Delta N_{{\rm eff},\gamma}$	$T_{\nu,1}$	$m_{\nu,1}^{(1)}$	$m_{\nu,1}^{(2)}$	$m_{\nu,1}^{(3)}$	$T_{\nu,2}$	$m_{\nu,2}^{(1)}$	$m_{\nu,2}^{(2)}$	$m_{\nu,2}^{(3)}$	$\cdots$

Table 1: Schematic form of the table through which

N

naturalness is implemented in CLASS. The

\Delta N_{\rm eff}^{\gamma}

encodes the contribution of the

N

-photon for all SM-like sectors. For each sector, the temperature and masses of the neutrinos are computed following Ref. Arkani-Hamed:2016rle. We assume a normal hierarchy for the neutrino mass, however, in CLASS we employ the degenerate neutrino approximation. The table is computed for a dense grid of

r

and

m_{\phi}

, and CLASS uses interpolation technique to calculate quantities for the in-between grid points. Two different tables are used for the Dirac and Majorana cases.

4 $N$ -neutrino Signals

In this section we consider the range of signals from the neutrinos of the additional SM-like sectors. The signals are distinct for the case of Dirac neutrinos and the case of Majorana neutrinos. While the temperature $T_{\nu,i}$ of the neutrinos of the $i$ th sector is independent of the nature of the neutrinos, the neutrino mass, on the other hand, has a different scaling. Dirac neutrinos have masses that scale as

m_{\nu,i}^{\rm Dirac}=\left(\frac{v_{i}}{v_{\rm SM}}\right)m_{\nu,{\rm SM}}\,,

(4)

where $v_{i}$ is the VEV of sector $i$ , $v_{\rm SM}$ is the VEV of the SM, and $m_{\nu,{\rm SM}}$ is the mass of the neutrinos in the SM. For Majorana neutrinos, the scaling is

m_{\nu,i}^{\rm Majorana}=\left(\frac{v_{i}}{v_{\rm SM}}\right)^{2}m_{\nu,{\rm SM}}\,.

(5)

We review the cosmological signatures of neutrinos in terms of the mass and temperature. We will study how these signatures change with sector and $N$ naturalness parameters using the scaling relations discussed in Sec. 2. We will also see that the signatures are vastly different for Dirac and Majorana neutrinos.

4.1 Matter Power Spectrum

ML: Self reminders: considering adding knr back into Eq 4.6 Change LSS to nl Understand features in triangle plots Update acknowledgements (PITT PACC grant number) Discuss how steps get smeared out due to R to NR transition Check usage of $m_{\nu,SM}$

Roughly speaking a given neutrino species will act as cold dark matter when non-relavistic and as radiation when relativistic. Interesting collective effects emerge due to the tower of neutrino species that spans a range of masses compared to the typical warm dark matter scenario with a single species. In particular, the matter power spectrum is sensitive to the times at which a neutrino species is relativistic or non-relavistic.

Figure 2 shows the ratio of the matter power spectrum in $N$ naturalness to the matter power spectrum in $\Lambda$ CDM as a function of wavenumber $k$ . The $N$ naturalness parameters used are $r=0.1$ , $m_{\phi}=160~{}{\rm GeV}$ , and $N_{\nu}=20$ and for the $\Lambda$ CDM case the value of $\Delta N_{\rm eff}$ and the total matter density $\omega_{m}$ is adjusted to match the values in the $N$ naturalness case. The neutrinos of the SM are taken to be triply-degenerate with a sum of masses of $0.12~{}{\rm eV}$ . The left plot shows the case of Dirac neutrinos and the right plot shows the case of Majorana neutrinos. In both cases there is a suppression of the matter power spectrum at small scales, but the scale at which the suppression starts occurring and the overall amount of suppression differs.

A cosmological case study of a tower of warm dark matter states: 𝑵Nnaturalness