Degenerate Fermion Dark Matter from a Broken $U(1)_{\rm B-L}$ Gauge Symmetry

The extension of the Standard model by assuming $U(1)_{\rm B-L}$ gauge symmetry is very well-motivated since it naturally explains the presence of heavy right-handed neutrinos required to account for the small active neutrino masses via the seesaw mechanism and thermal leptogenesis. Traditionally, we introduce three right handed neutrinos to cancel the $[U(1)_{\rm B-L}]^3$ anomaly. However, it suffices to introduce two heavy right-handed neutrinos for these purposes and therefore we can replace one right-handed neutrino by new chiral fermions to cancel the $U(1)_{\rm B-L}$ gauge anomaly. Then, one of the chiral fermions can naturally play a role of a dark matter candidate. In this paper, we demonstrate how this framework produces a dark matter candidate which can address the so-called"core-cusp problem". As one of the small scale problems that $\Lambda$CDM paradigm encounters, it may imply an important clue for a nature of dark matter. One of resolutions among many is hypothesizing that sub-keV fermion dark matter halos in dwarf spheroidal galaxies are in (quasi) degenerate configuration. We show how the degenerate sub-keV fermion dark matter candidate can be non-thermally originated in our model and thus can be consistent with Lyman-$\alpha$ forest observation. Thereby, the small neutrino mass, baryon asymmetry, and the sub-keV dark matter become consequences of the broken B-L gauge symmetry.


Introduction
Despite various evidences for the presence of dark matter (DM), DM's nature has not been uncovered yet. The central questions in regard to DM concern a mass of DM, non-gravitational interaction DM does, and its stability. Answers to these questions are considered essential factors in understanding not only a history and structure of the Universe in cosmology, but also a bigger and more fundamental picture lying behind the Standard model (SM) in particle physics. Seen from the perspective of this kind, DM related-observational anomalies reported in the study of cosmology and astrophysics could serve as a critical hint for physics beyond the Standard Model (BSM) although it is not necessary.
With that being said, a well known discrepancy between what has been expected based on a standard hypothesis of the cold dark matter (CDM) and what are observed regarding the small scale structure (galactic or sub-galactic scale) may deserve attention from a well-motivated BSM physics. Cuspy halo profiles of dwarf galaxies predicted by N-body simulations equipped with CDM [1,2,3] are at odds with the cored halo profiles implied by stellar kinematic data of low mass galaxies [4,5,6,7,8], which might be signaling a nature of DM deviating from collisionless and cold aspects. Along with "the missing satellite problem" [9,10] and "too-big-to-fail problem" [11], this socalled "core-cusp problem" [12] is challenging for the most popular and robust CDM framework in spite of the success it has achieved thus far in accounting for the large scale structure (LSS) of the Universe and evolution thereof.
Relying on predicted phenomenological consequences arising from a specific mass or non-gravitational interaction that DM enjoys, several alternative frameworks to CDM have been suggested so far in an effort to address the core-cusp problem (and other small-scale issues as well). These include warm dark matter (WDM) 1 [13,14,21] , ultra-light bosonic DM [22,23,24] 2 and self-interacting DM (SIDM) 3 [27]. 1 Warm dark matter is assuming a DM particle characterized by a small enough mass to produce a freestreaming length of O(0.1)Mpc and also by non-zero velocity dispersion. This feature enables WDM to erase density perturbations for the scale smaller than its free-streaming length, and thereby suppression of matter power spectrum on small scale and of formation of sub-halos is induced in comparison to CDM case [13,14,15,16,17,18] . As for the core size of a dwarf galaxy, WDM N-body simulations were conducted to study how the primordial velocity dispersion of WDM affects the inner structure of DM halo in [19,20]. In particular, sub-keV WDM is shown to produce the halo core size of O(100)pc for a typical sub-halo mass of Milky way whereas WDM with 1 − 2keV mass does the core of 10-50pc [20]. 2 The quantum pressure of the ultra-light bosonic dark matter supported by Heisenberg's uncertainty principle can help the self-gravitating system achieve stability against gravitational collapse. 3 The self-interaction helps efficient heat conduction from the outer more energetic DM particles to the inner colder ones, which leads to redistribution of energy and angular momentum of DM particles. Consequently, as was shown in relevant simulations [25,26] , the central halo becomes less dense compared Another interesting possibility of DM resulting in a cored halo profile in a low mass galaxy is the fermion DM in the quantum degenerate limit. Along the similar line, the hypothesis of the fermion DM as the self-gravitating (quasi) degenerate gas was invoked in [28,29,30,31,32] to explain the kinematics of dwarf spheroidal galaxies (dSphs). The fitting procedures for the kinematic data (stellar velocity dispersion and halo radius of dSphs) yielded sub-keV mass regime as the possible fermion DM mass (See also Refs. [33,34,35,36] for more studies about sub-keV fermion DM). Because of this, sub-keV fermion DM can serve as a class of solution to the core-cusp problem if it resides in dSphs nowadays with sufficiently low temperature so as to sit in the (quasi) degenerate state. This sub-keV mass regime encounters a severe constraint from the Lyman-α forest (see, e.g. [37,38,39]), but the solution can still be viable provided that the sub-keV fermion DM is non-thermally originated and the free-streaming length of DM is not too large to be consistent with constraints derived from the Lyman-α flux power spectrum. The free-streaming length range 0.3Mpc < λ FS < 0.5Mpc of DM would be of interest since it can be consistent with non-vanishing matter power spectrum at large scales and avoid too many satellites of Milky way size halo [40,41,14,28].
Given the problem and one of the answers to it described above, the next question naturally thrown from the particle physics side could be probably whether a wellmotivated extension of SM can accommodate such a sub-keV degenerate fermion WDM. In this work, we give our special attention to an extension of SM with a gauged U (1) B−L symmetry. On top of SM particle contents, in its minimal form, the model contains two heavy right handed neutrinos (N i=1,2 ) and a complex scalar (Φ) for breaking U (1) B−L . By means of this basic setting, the model is expected to accomplish the successful explanation for the small neutrino masses via seesaw mechanism [42,43,44] and the baryon asymmetry via the thermal leptogenesis [45]. Now for the purpose of making the theory anomaly-free and accommodating a non-thermal sub-keV fermion DM candidate, more chiral fermions are added to the model. The similar framework was studied in Refs. [46,47,48] under the name of "Number Theory Dark Matter".
Beginning with the minimal set of new U (1) B−L charged particle contents relative to SM, we shall search for all the possible sub-keV fermion DM production mechanisms and check consistency with Lyman-α forest observation. We gradually move to the next minimal scenario whenever an inconsistency is detected. Finally, we arrive at scenarios where multiple fundamental questions of small neutrino masses, baryon asymmetry and DM resolving small scale could be dealt with at once. We shall test consistency with Lyman-α forest data by computing free-streaming length of DM and constructing a map between thermal WDM mass and our sub-keV fermion DM mass. As an additional to CDM case and a cored halo profile forms accordingly.
consistency check, we compute ∆N BBN eff contributed by the sub-keV fermion DM and "would-be" temperature today of the DM candidate.

Model
As the starting point of the task to extend the SM in a minimal way, we introduce a gauged U (1) B−L symmetry. As the most elegant way of explaining the small neutrino masses, seesaw mechanism predicts the presence of heavy right-handed neutrinos [42,43,44]. The advantage of extending the gauge symmetry group of SM by including the gauged U (1) B−L lies in precisely this point. The theory can be naturally rendered gauge anomaly free when there exist three right-handed neutrinos with the opposite lepton number to that of active neutrinos. The other remarkable consequence that immediately follows here is that the presence of the heavy right-handed neutrinos can help us understand imbalance between baryon and anti-baryon abundance. Induced by the out-of-equilibrium decay of the right-handed neutrino, the primordial lepton asymmetry can be converted into baryon asymmetry by sphaleron transition [45].
Motivated by these attractive aspects, we consider a variant of SM with the gauge Concerning the particle contents of the model, we begin with SM particle contents plus only two right-handed neutrinos, which is the most economical addition for the seesaw mechanism and the successful thermal leptogenesis [49]. In addition, we introduce one complex scalar to the model of which vacuum expectation value (VEV) causes the spontaneous breaking of U (1) B−L . Via Majorana Yukawa coupling, the complex scalar imposes masses to the two right handed neutrinos on U (1) B−L breaking. Of course, one is naturally tempted to make introduction of three right-handed neutrinos at the moment since it can satisfy the anomaly free condition of U (1) B−L and simultaneously may be able to explain DM by taking the lightest right handed neutrino (sterile neutrino) as a candidate of the dark matter. 4 However, there is no natural and convincing reason for such a large mass disparity between the first two and the last right-handed neutrinos. Therefore, assuming only two right-handed neutrinos for the seesaw mechanism and the thermal leptogenesis, we need to find another way to accommodate DM candidate in the model. For this purpose, recalling the necessity for making the model U (1) B−L anomaly free is of a great help. What could help to render U (1) B−L anomaly free on behalf of the third right-handed neutrino? Looking at the anomaly free condition for U (1) B−L given in Eq. (1) below, 5 where the sum is over fermions charged under U (1) B−L , and referring to Table 1, one comes to realize that a new set of fermions to be added to the model should satisfy where the sum is over a new set of fermions. Here the solutions including vectorlike fermions are out of our interest. Probing the cases to meet the two conditions in Eq.
(2) simultaneously leads us to the conclusion that minimum number of the new chiral fermions to be added is four. 6 In effect, the similar logic was studied in [46,47] and diverse combinations of possible Q B−L values were found there. Given many options, for our work, we choose Q B−L assignments shown in Table 2 for the new chiral fermions.
ψ −9 ψ −5 ψ 7 ψ 8 Q B−L -9 -5 +7 +8 Now the U (1) B−L charge assignment given in Table 1 and 2 leads to the following renormalizable Yukawa couplings between Φ −2 and fermions in the model charged 5 The first one is for cancellation of U (1) 3 B−L anomaly and the second one is for cancellation of gravitational U (1) B−L × [gravity] 2 anomaly. The anomalies of U (1) B−L ×[SM gauge interactions] 2 are canceled by the quark-sector contributions. 6 Due to the Fermat's theorem, solutions with two additional Weyl fields do not exist. Solutions with three extra Weyl fields always contain two vector-like fermions.
Once U (1) B−L is spontaneously broken by acquisition of VEV (<Φ −2 >≡ V B−L ) of Φ −2 , there arise mass eigenstates ψ 7 , χ and ξ with χ ≡ y 1 Here χ and ψ 7 form a Dirac fermion under which all fermions except for ψ 8 are odd. Thus, ψ 8 is perfectly stable. In accordance with this observation, we take ψ 8 as the DM candidate in our model. We attribute the stability of DM to even Q B−L of ψ 8 from now on. ψ 8 obtains its mass via a higher dimensional operator where κ is a dimensionless coefficient. Here we see that U (1) B−L breaking scale directly determine the DM mass.
To estimate the mass of ξ, we can try to figure out the smallest mass eigenvalue in terms of V B−L /M P by writing down 4×4 mass matrix for the fermion field vector F ≡ (ψ −9 , ψ −5 , ψ 7 , N (i) ) formed by not only renormalizable, but higher dimensional operators consistent with assumed symmetries. Particularly, owing to the terms 7 the mass of ξ is given by GeV .
Interestingly, we obtained the small mass of the dark matter in the sub-keV regime, for V B−L 2 × 10 15 GeV owing to its B − L charge. Then, m ξ becomes as large as ∼ O(10 9 )GeV.

Sub-keV Fermion Dark Matter Production from Scattering of SM Particles?: Maybe Not
In the previous section, we discussed a well-motivated minimal extension of SM in which sub-keV fermion DM may arise. As was pointed out in the introduction, we go through the procedure to check cosmological history and free-streaming length of DM candidate in the model in order to see whether the minimal model is good enough to be consistent with cosmological constraints including Lyman-α forest data. If not, we would gradually move to a next-to-minimal model by enlarging particle contents. From this place, since we are interested in sub-keV DM mass regime, we assume V B−L (2 − 3) × 10 15 GeV based on Eq. (6), which is also consistent with the observed neutrino masses. In the minimal model, the only particle that is communicating with ψ 8 at the renormalizable level is U (1) B−L gauge boson (A µ ). Thus, the only way for ψ 8 to be produced is pair production resulting from scattering among the SM particles via the virtual U (1) B−L gauge boson exchange. 8 The corresponding Feynman diagram is shown in Fig 1. This production, however, should proceed with Γ(SM + SM → ψ 8 + ψ 8 ) < H at the reheating era. Otherwise, ψ 8 is thermalized by the SM thermal bath to become the thermal WDM which cannot have sub-keV mass regime. Thus we require For this production route which is most efficient at the reheating era, with the assumption that ψ 8 is identified as the sole DM component today, the DM number density to entropy density ratio reads [52] where n f SM ∼ T 3 is the SM fermion number density, Γ ≡ n f SM <σv> is the interaction rate for scattering among SM fermions, g * is the effective number of relativistic degrees of freedom and s SM = 2π 2 g * T 3 /45 is the entropy density. We assume the mass of A µ is greater than a reheating temperature so that A µ is never present in the SM thermal bath. Now the use of Eq. (11) and Y DM ≡ n DM /s SM 4.07 × 10 −4 × (m DM /1keV) −1 along with Eq. (6) above yields 9 3.9 × 10 −3 g * 100 In other words, for given a m DM , T RH required for production of the correct amount of DM abundance today via scattering among SM fermions must be GeV . 9 From Ω DM,0 = 0.24, H 0 = 70km/sec/Mpc and s SM,0 2.945 × 10 −11 eV 3 (entropy density today), DM abundance Y DM ≡ n DM /s SM is expressed in terms of DM mass as at DM production time.
For g * 100, m DM O(100)eV and κ O(1), T RH in Eq. (14) reads ∼ O(10 13 )GeV. Provided that the right amount of DM is produced at the reheating era with this reheating temperature, we also expect production of ξ via the similar SM scattering with the production ratio On production, we anticipate that ξ completes decaying to SM Higgs and lepton before EW symmetry breaking time is reached and so it is cosmologically harmless (for detail, see Appendix A). Due to the small interaction rate, ψ 8 starts free-streaming since production near the reheating era. The free-streaming length must be checked to be at least smaller than 0.5Mpc. This is for avoiding too much suppression of the matter power spectrum on small scales inconsistent with observation. The free-streaming length of DM is computed by where t p and a p are the time and the scale factor at which DM starts free-streaming, < v DM (t) > is the average velocity of the dark matter at the time t, and < p DM (a p ) > is the DM momentum at t p . Ω rad,0 and Ω m,0 denote the radiation and matter density parameters, respectively. Even if ψ 8 does not form a dark thermal bath, its momentum distribution is expected to be similar to the thermal distribution since it is produced from scattering of SM fermions which are in the thermal bath. The average momentum of the DM is estimated as where a p = a RH (10 −13 GeV)/T RH is the time of the onset of DM free-streaming. 10 The factor 3.15 applies for the typical thermal distribution of fermions. Using Eq. (17), estimation of λ FS for even m DM = 1keV yields 1.25Mpc. The smaller DM mass corresponds to the longer λ FS than this. This estimation concludes that the minimal scenario cannot produce a degenerate sub-keV fermion DM candidate for explaining the cored DM profiles for dSphs.
Then, what another way could be considered to produce sub-keV fermion DM with a shorter free-streaming length? We notice that decreasing T RH cannot shorten λ FS in Eq. (16) as long as T RH 10MeV where 10MeV is the lower bound of T RH from BBN. On the other hand, because ψ 8 cannot be coupled to any particle in the model other than U (1) B−L gauge boson at the renormalizable level, 11 no other DM production mechanism can be envisioned in the minimal model. Hence we cannot help but conclude that λ FS cannot be shorten unless another DM production mechanism is considered by modifying the minimal model. Therefore, we conclude that ψ 8 produced from SM particle scattering cannot be a candidate for the degenerate sub-keV DM to resolve the core-cusp problem. Now the whole of reasoning we followed in Sec. 3 necessitates searching for a new way of producing DM which we discuss in the next section.

Sub-keV Fermion DM from Inflaton Decay
As a next step, let us consider the DM production from the inflaton decay. ψ 8 can be coupled to the inflaton via where Φ I is the inflaton field and is assumed to be a gauge singlet and Lorentz scalar from here on. If the mass of the B-L gauge boson, m B−L , is larger than the inflaton mass, i.e. m B−L > m I , the decay rate of the process Φ I → ψ 8 +ψ † 8 +B-L charged particles (X) is To explain the current abundance of the dark matter, one requires where Br is the branching ratio of Φ I → ψ 8 + ψ † 8 + X to the inflaton decay rate. 12 The  Figure 2: The ratio of a reheating temperature (T RH ) to an inflaton mass (m I ) that results in DM's free-streaming length 0.3Mpc < λ FS < 0.5Mpc when DM is directly produced from the inflaton decay.
reheating temperature is To avoid the dominant production from the SM thermal bath (the case discussed in Sec. 3), T RH 10 13 GeV is required. In addition, the ratio of a reheating temperature (T RH ) to an inflaton mass (m I ) that results in DM's free-streaming length 0.3Mpc < λ FS < 0.5Mpc is O(0.1) − O(1) as can be seen in Fig. 2 when DM is directly produced from the inflaton decay with <p DM (a p )>= m I /2 and a p = a RH . However, this ratio with Eq. (21) for T RH 10 13 GeV leads to the condition This is inconsistent with Fig. 2. Thus, this possibility is out of our interest. On the other hand, for m B−L < m I , the decay rate is where g B−L denotes the B-L gauge coupling. To explain the dark matter density, we require This leads to Similar to Eq. (21), Eq. (25) gives rise to which is inconsistent with Fig. 2. Thus, this possibility is also out of our interest. 13 Therefore, we need to extend the minimal model to have the degenerate fermion DM. As we will explain in detail, one simple possibility is to introduce a complex scalar field Φ 16 with Q B−L = 16. This scalar field couples to DM through where y * is a dimensionless coupling. 14 The renormalizable scalar sector potential we consider in the following reads 15 where m 16 is a parameter with a mass dimension, λ 16 is a dimensionless coupling, and 13 Along with Eq. (26), too large a mass value itself for the inflaton also makes ψ 8 production from the inflaton decay with m B−L < m I not viable. From 14 Similar to ψ 8 , Φ 16 could be produced from SM particle scattering as long as T RH is large enough. The relevant diagram would be the one in Fig. 1 with ψ 8 replaced by Φ 16 . For this route, due to Q B−L ratio, we expect four times more production of Φ 16 than that of ψ 8 . This case is also out of our interest because significant amount of DM (∼ 25%) would travel too large a free-streaming length as shown above using Eq. (16) and (17). 15 For the purpose of preventing Φ 16 from being thermalized by any particle, we assume sufficiently suppressed renormalizable mixing of Φ 16 with other scalars like ∼ (H † H)|Φ 16 | 2 and ∼ |Φ I | 2 |Φ 16 | 2 which are allowed by symmetries in the model. See appendix. B for more discussion about the Higgs portal couplings.
g is a parameter with a mass dimension, V (Φ I ) and V (H) are the potential for inflaton and SM Higgs doublet. We take < Φ 16 >= 0 in the vacuum, assuming the Φ 16 has a positive mass squared, m 2 16 > 0. This makes Eq. (6) intact. In Sec. 4, we assume that the Hubble induced mass squared for the Φ 16 is positive so that Φ 16 sits near the origin of the field space during and in the end of inflation. Now Φ 16 may be produced from the inflaton decay at the reheating era via the decay operator ∼ gΦ I |Φ 16 | 2 if m I 2m 16 holds. In this section, we attend to Φ 16 particle produced in this manner. We are aiming to show that such a Φ 16 could be a mother particle producing sub-keV fermion DM (ψ 8 ) consistent with Lyman-α forest observation. Depending on a value of λ 16 , we have two different scenarios. We explore a case where a dark sector thermal bath forms in Sec. 4.1 and the other case where a dark sector thermal bath never forms in Sec. 4.2.

The Case with Formation of Dark Sector Thermal Bath
In this section, we consider the case in which a dark thermal bath purely made up of Φ 16 forms when Φ 16 is produced from the inflaton decay. When λ 16 = 0 holds, from the comparison where T D is the temperature in the dark sector, we realize that it is easy for a dark thermal bath made up of Φ 16 to form as far as the quartic interaction (λ 16 ) of Φ 16 is not too small. Here x = T D /T SM is a fraction of order O(0.1) to be determined by DM relic density matching. We define the branching ratio Br to satisfy n 16 = Br × n I Br × (ρ SM /m I ) 16 where n 16 and n I are the number density of Φ 16 and inflaton (Φ I ) respectively. We assume ρ I ρ SM at the reheating era. From the number density comparison, we obtain the relation between dark sector temperature and SM sector 16 This relation n 16 Br × (ρ SM /m I ) can be used to derive relation between m I , T RH and Br. Using the approximation 2n φ16 = n DM at production time, one obtains Using Eq. (12) and (30), one obtains temperature where we used g D (a RH ) = 2 and g SM (a RH ) = 106.75. The second equality is coming from Eq. (31). The ratio of T D /T SM remains the same until Φ 16 decays to a DM pair. We note that Br is lower bounded as Br (2.7 × 10 −4 (m DM /1keV) −1 ) 2 . This constraint is derived from the condition that Φ 16 never gets into the SM thermal bath by the decay and the inverse decay process of Φ I ↔ Φ 16 + Φ * 16 , and the requirement of obtaining the correct DM density (see Eq. (31)). 17 Concretely, we consider a scenario in which Φ 16 becomes non-relativistic in the dark thermal bath before the time of Γ(Φ 16 H is reached. Afterwards, nonrelativistic Φ 16 decays to DM pair when the time of Γ(Φ 16 → ψ 8 + ψ 8 ) H is reached. The similar scenario was considered in [48]. We demand that DM does not exist at the reheating era and is produced only from the decay of Φ 16 . To this end, define T SM,i (T D,i ) to be the SM (dark) thermal bath temperature at which Γ i H holds. For Φ 16 + Φ 16 scattering to produce DM+DM and vice versa via t-channel DM exchange shown in Fig. 3, the interaction rate reads, From the first inequality in Eq. (35), we obtain 17 The condition is Br × T 2 RH /M P m 2 I /M P where the process Φ I ← Φ 16 + Φ * 16 is ineffective until the inflaton becomes non-relativistic and disappears.
In addition, requiring that DMs do not form a thermal bath via their self-interaction through Φ 16 exchanges after its production Thus, we can see that for m 16 10 13 GeV, Eq. (37) and (39) are satisfied as long as Eq. (36) is so. Together with m 16 , y * is treated as a free parameter as far as y * y * ,max is satisfied. The smaller y * becomes, the later time onset of the free-streaming of DM becomes. Given a fixed initial momentum <p DM (a FS )> m 16 /2, the larger a FS implies a larger <p DM (a > a FS )> for a fixed scale factor a > a FS . In the light of the fact that the late universe contribution to λ FS is greater than the earlier one, we are led to speculate that for the same (m 16 , m DM ), the smaller y * would lead to the larger λ FS and hence more stringent constraint on m DM . To constrain the model, we consider the free-streaming length criterion 0.3Mpc < λ FS < 0.5Mpc. Following Eq.
where <p DM (a FS )> m 16 /2 was used with a p a FS . Here a p and a FS are the scale factor at which the production of DM and the free-streaming of DM take place, respectively. Using Eq. (34), a p can be computed via For a fixed (m 16 , m DM ), y * max in Eq. (37) is determined, defining an allowed range of y * < y * max . Within the range, the smaller y * results in the longer λ FS since the freestreaming is delayed with the same initial momentum <p DM (a FS )> m 16 /2. This means that for each set of (m 16 , m DM ), y * = y * ,max in Eq. (41) yields the smallest λ FS value. On the other hand, for y * = y * ,max , we notice that λ FS in Eq. (40) becomes independent of m 16 since m 16 a p is so. Thus, we realize that for y * = y * ,max , λ FS is minimized for each m DM whatever m 16 is. In Fig. 4, we show λ FS computed with y * = y * ,max for the dark matter mass range 0.1keV m DM 1keV. For a smaller y * choice, the curve in Fig. 4 would move upward. Without going through the further study with y * smaller than y * ,max , we restrict ourselves to the case with y * = y * ,max as an example, but the logic presented below can be also applied to other values of (y * , m 16 ) for the consistency check.
Starting with the momentum m 16 /2 at a = a FS , the sub-keV DM we discuss here is still relativistic at BBN era with the momentum ∼ O(1)MeV. As such, the sub-keV DM serves as an extra-radiation during BBN era and therefore its contribution to ∆N BBN eff needs to be checked to be consistent with the known constraint. For each m DM , we computed ∆N BBN eff contributed by DM at the BBN era and found that the model with y * = y * ,max is consistent with ∆N BBN eff 0.114 (95% C.L.) recently reported in [54]. For computation of ∆N BBN eff contributed by DM, we refer the readers to Appendix C. As the final consistency check, we estimated the "would-be" temperature today (T DM,0 ) for ψ 8 based on Eq. (72) which reads where we used y * = y * ,max in Eq. This confirms that the current temperature of DM becomes low enough to accomplish degenerate configuration when structure formation is ignored. We notice that y * can be constrained by ∆N BBN eff , which we do not explore in detail. Intriguingly, for y * = y * ,max , the criterion 0.3Mpc λ FS 0.5Mpc gives the mass constraint 0.25keV m DM 0.37keV, which lies in the range of degenerate fermion DM mass accounting for the cored DM profiles of dSphs in Refs. [28,29,30,31]. Another choice of y * < y * ,max will make 0.3Mpc λ FS 0.5Mpc correspond to a larger m DM range.
Additionally, we also discuss the constraint on the mass of our DM candidate (ψ 8 ) mapped from a conservative lower bound for the mass of the thermal WDM, i.e. 1.9keV (95% C.L.), recently reported in [55]. We make a detail discussion about how the mapping can be achieved in Appendix D. Here we directly construct the map based on Eq. (69). We begin by equating the warmness parameters for ψ 8 (σ ψ 8 ) and the thermal WDM (σ wdm ) where m and T denote a mass and temperature, andσ is defined in Eq. (68). As a particle produced from the decay of a non-relativistic mother particle, ψ 8 is characterized by the momentum space distribution function f (q, t) = (β/q)exp(−q 2 ) where β is a normalization factor and q ≡ p/T is used [56,57,58,59,60]. This gives usσ ψ 8 1. On the other hand, since m 16 >> m DM is assumed, DM temperature at the matter-radiation equality can be written as with a FS defined in Eq. (41). Finally, by usingσ wdm = 3.6 for the thermal WDM and T wdm (a eq ) = T wdm,0 /a eq in Eq. (70), we obtain the map Applying the conservative constraint m wdm > 1.9keV [55], we obtain m ψ 8 0.4keV. This result may seem a tension with m DM required for a degenerate fermion DM in [28,29]. However, indeed there exist some uncertainties in velocity anisotropy parameter used for fitting of the stellar velocity dispersion, the lower bound of Fornax dSphs halo radius and baryon's effect on the DM halo profile. Also still for some dSphs other than Fornax, the best fitting for the stellar velocity dispersion is done by m DM as large as 550-650eV [30]. Here without performing a detailed fitting analysis to infer the degenerate fermion DM mass, we take a conservative attitude to understand 100eV m DM 1keV as the interesting range relating to degenerate fermion DM solution to the core-cusp problem.

The Case without Formation of Dark Sector Thermal Bath
For the case where Φ 16 does have a tiny or vanishing quartic interaction, Φ 16 would not form a dark thermal bath as far as Yukawa interaction with ψ 8 is sufficiently small. Since production from the inflaton decay, it would continue to free-stream until it decays to a pair of ψ 8 . Note that this early free-streaming of Φ 16 is not problematic at all for the small scale perturbations since the early time free-streaming length is negligibly small. With this picture in mind, in this section, we study the possibility of having degenerate fermion DM arising from the decay of a free non-relativistic scalar Φ 16 . We explore the parameter space of the model where the free-streaming length of ψ 8 becomes consistent with Lyman-α forest observation.
In order to avoid having the thermal WDM, we focus on the scenario where Φ 16 starts the free-streaming once produced from the inflaton decay. After that Φ 16 becomes non-relativistic first and then decays to DM pairs. Differing from the previous case with λ 16 = 0, the time when Φ 16 becomes non-relativistic is sensitive to inflaton mass now. Φ 16 has momentum p 16 (a RH ) m I /2 at the reheating era on production and then becomes non-relativistic at where we used a RH (10 −13 GeV)/T RH for the third equality and Eq. (31) for the last equality. For our purpose, we demand that where T SM,1 and T SM,2 were defined in Eq. (33) and Eq. (34). From the first inequality in Eq. (47), we obtain From the second inequality in Eq. (47), we obtain In addition, as discussed in Sec. 4.1, we require so that the DM does not form a dark thermal bath via their self-interaction through Φ 16 exchange after its production. For a given set of (m 16 , m DM , Br), each of (y * ,1 , y * ,2 , y * ,3 ) is to be determined. Define y * ,max ≡ min(y * ,1 , y * ,2 , y * ,3 ). Then a choice of Yukawa coupling satisfying y * < y * ,max will satisfy Eq. (47). Numerically we find that (1) for Br 10 −3 , y * ,max = y * ,1 for any sub-keV m DM and (2) for Br 10 −4 , y * ,max is either y * ,1 or y * ,3 . For a fixed m DM , λ FS depends on m 16 and y * , and these two are inversely-correlated. Thus, in principle, for a fixed m DM , a set of (m 16 , y * ) satisfying λ FS ∈ (0.3, 0.5)Mpc can be readily found and consistent insofar as y * < y * ,max . In this section, instead of probing all the allowed parameter space for (m 16 , Br, y * , m DM ), for our purpose it suffices to choose a specific benchmark set of parameters (m 16 = 5 × 10 5 GeV, Br = 10 −6 , y * = 5 × 10 −6 ) to show that a degenerate sub-keV fermion DM can be produced in the model. Then we see that y * < y * ,max is satisfied. We emphasize that this example is not atypical and the following logic and consistency check can also apply for other values of parameters. The result of computation for λ FS (m DM ) is shown in Fig. 5. Interestingly, the range 0.2keV m DM 0.35keV corresponds to the criterion 0.3Mpc λ FS 0.5Mpc gives the mass constraint , which lies in the range of degenerate fermion DM mass accounting for the cored DM profiles of dSphs in Refs. [28,29,30,31]. The smaller y * and the larger m 16 would make the curve in Fig. 5 move upward.
As the final consistency check, we compute ∆N BBN eff and "would-be" temperature today for ψ 8 . Firstly, from Eq.

Sub-keV Fermion DM from Decay of a Scalar Field Coherent Oscillation
So far we have assumed that Φ 16 has a positive Hubble induced mass squared during the inflation. However, we assume the negative Hubble induced mass squared in this section. We consider the potential of Φ 16 , where H inf is the Hubble parameter during inflation, n is a positive integer larger than one, c 2 and c 2n are positive dimensionless couplings. Then, Φ 16 sits around the potential minimum with the amplitude Φ 16,I during the inflation, where we ignore the mass term with m 16 by assuming m 2 16 c 2 H 2 inf . After the end of inflation, the field value of Φ 16 is given by for n ≥ 4. Here, H denotes the Hubble expansion rate. This behavior of the scalar field is called the scaling solution [61,62,63]. We focus on this scaling solution with n = 4 as an example in the rest of this section. 18 As the Hubble expansion rate decreases and when it becomes comparable to m 16 , the scalar field Φ 16 starts the coherent oscillation around its origin. After that, when Γ(Φ 16 → ψ 8 + ψ 8 ) H holds, Φ 16 decays into the DMs. 19 This mechanism is basically the same as the one discussed in Sec. 4 whereas Φ 16 production mechanism is different.
In the above DM production from the coherently oscillating Φ 16 , the abundance of 18 We ignore the other terms with n = 4 not to affect the dynamics of Φ 16 . The analysis for the potential with n = 2 or 3 will be given elsewhere. 19 Regarding the constraint from the isocurvature perturbations, the fluctuation of Φ 16 is imprinted in the DMs in our mechanism. Thus, we assume c 2 O(10) to suppress the isocurvature perturbations (see e.g. Ref. [64]). Note that the fluctuation of the axial component of Φ 16 is not suppressed by this way, but this does not matter because only the fluctuation of the radial component of Φ 16 leads to the isocurvature perturbations of the DM.
where a osc is the scale factor as the oscillation starts. Notice that we assumed that the oscillation starts at the radiation-dominated era (T osc < T R ). By attributing the whole current DM abundance to ψ 8 , we demand 2n 16 /s SM = Y DM at a = a osc which yields g * (a osc ) 100  Fig. 6, requiring 0.3Mpc < λ FS < 0.5Mpc constrains the space of the Yukawa coupling between DM and Φ 16 . m 16 . As an example, for c 2 /c 8 = 1, 5, 10, we show this map in Fig. 6.
After the right amount of Φ 16 is generated, in order to have ψ 8 as a degenerate fermion DM candidate today, we demand that where T SM,2 was defined in Eq. (34). This leads to y * < 2.7 × g * (a osc ) 100 In addition, as discussed in Sec. 4.1, we require so that the DMs do not form thermal bath via their self-interaction through Φ 16 exchanges after its production. Define y * ,max ≡ min(y * ,1 , y * ,3 ). Now for a set of (m 16 , m DM , c 2 /c 8 ) satisfying Eq. (57), y * ,max is determined, and by choosing a y * y * ,max , λ FS can be computed based on Eq. (40) and required to be 0.3Mpc < λ FS < 0.5Mpc. For an example of c 2 /c 8 = 5, we go through this procedure to constrain the space of the Yukawa coupling between DM and Φ 16 , of which the result is shown in Fig. 7. For this y * , it turns out that Φ 16 decay takes place before BBN era (a p O(10 −15 ) − O(10 −11 )) and therefore sub-keV DM contributes to ∆N BBN eff . Based on Eq. (64), we compute ∆N BBN eff attributable to DM and find it is at most ∼ 0.01 to be consistent with ∆N BBN eff 0.114 (95% C.L.) [54]. From Eq. (72), we also estimate DM's "would-be" temperature today for the case with c 2 /c 8 = 5. The results readT DM,0 ∼ O(10 −8 ) − O(10 −7 )K which is smaller than This shows that current temperature of DM becomes low enough to accomplish the degenerate configuration when structure formation is ignored. We do not go further to discuss the cases with different ratios of c 2 /c 8 . If one findsT DM,0 > T DEG , one may arrive at a value of c 2 /c 8 which is not allowed. But we note that m 16 and a FS are inversely correlated in Eq. (72).

Conclusion
In this paper, we present a well-motivated extension of the SM which can address the core-cusp problem by providing a degenerate sub-keV fermion DM candidate. The model is characterized by U (1) B−L gauge symmetry, and two right-handed heavy neutrinos and four new chiral fermions added to the SM gauge sector and particle contents respectively. All the fermions in the model are charged under U (1) B−L and assigned the corresponding Q B−L s in a way that U (1) B−L is rendered anomaly-free. It is extremely remarkable that one of the additional fermions obtains naturally a mass of O(1)keV because of its large B-L charge, provided that the B-L symmetry breaking scale ∼ 10 15 GeV. Thus, it was shown that the chiral fermion can serve as a sub-keV fermion DM candidate of which temperature today is low enough to form a degenerate fermion halo core for a dSphs. The DM's free-streaming length is small enough to be consistent with Lyman-α forest data. Being WDM, the DM candidate in the model is also expected to resolve other small scale problems that ΛCDM paradigm confronts (the missing satellite and too-big-to-fail problem). Consequently, the model can resolve the small scale issues in cosmology as well as the smallness of the active neutrino mass and the baryon asymmetry via the thermal leptogenesis.
Concerning the DM production mechanism, we argue that fermion DM produced from the decay of a complex scalar can meet the criteria for a degenerate fermion DM. In Sec. 3, we showed that non-thermal DM produced from the SM particle scattering is bound to travel too large a free-streaming length. In Sec. 4, we showed that DM produced from a series of decays (inflaton decay and Φ 16 decay) as the final product can travel the right size of the free-streaming length ∼ O(0.1)Mpc to be consistent with Lyman-α forest observation. Getting into more detail, we conducted the case study depending on whether a dark thermal bath forms (Sec. 4.1) or not (Sec. 4.2). For both cases, λ FS for a fixed m DM are parametrized by (m 16 , y * ). We figure out that for a set of (m 16 , m DM ), the constraint applied to a choice of y * is more stringent for the case with formation of a dark thermal bath (Sec. 4.1) than the other case (Sec. 4.2). This fact makes it easier for the case without a dark thermal bath to produce a degenerate fermion DM consistent with the free-streaming length criterion. In Sec. 5, we studied a different mechanism to produce the degenerate fermion DM via the decay of a scalar field coherent oscillation. Differing from Sec. 4 where a positive Hubble induced mass is assumed during inflation, a negative Hubble induced mass during inflation is assumed in Sec. 5. We studied a potential of Φ 16 in Eq. (51) by which Φ 16 field is located away from the origin in the field space at the end of the inflation. For a fixed c 2 /c 8 , there is one to one map between m DM and m 16 , which is required by DM relic density matching. Taking, for example, c 2 /c 8 = 5, we showed that how the free-streaming length criterion 0.3Mpc λ FS 0.5Mpc can constrain Yukawa coupling between the mother scalar field with ∼ m 16 ∈ (10 −3 , 10 3 )GeV and DM candidate. For all distinct DM production mechanisms, we also performed further consistency checks including ∆N BBN eff contributed by DM andT DM,0 < T DEG . Finally, we note that the framework presented in this paper shows that even if fermion warm DM mass is as low as sub-keV regime, it can still travel the free-streaming length as short as ∼ O(0.1)Mpc consistent with Lyman-α forest observation thanks to the non-trivial dark sector structure and its cosmological history.
Higgs portal is greatly suppressed. This tells us that produced from scattering among SM Higgs, Φ 16 would be easily thermalized by SM thermal bath with significant λ * . Once Φ 16 joins the SM thermal bath, trivially it never decouples. For the case where Φ 16 decays before Φ 16 becomes non-relativistic, ψ 8 becomes thermal WDM 20 which is out of our interest. On the contrary, if Φ 16 becomes non-relativistic before its decay to the DM starts, Φ 16 would disappear prior to production of ψ 8 . 21 For these reasons, for the purpose of having sub-keV non-thermal fermion WDM, it is necessary for us to assume a highly suppressed Higgs portal operator ∼ λ * (H † H)|Φ 16 | 2 .

C ∆N eff contributed by DM (ψ 8 )
Recalling the expression for the radiation energy density ρ rad (T 1MeV) ρ γ 1 + 7 8 4 11 we compute the extra-contribution to radiation from the relativistic DM at BBN era by where based on Eq.  20 The abundance of the WDM will be larger than the current dark matter abundance. 21 If Φ 16 is heavier than EW symmetry breaking scale, it will be Boltzmann suppressed once T SM m 16 is reached. If it is lighter than EW symmetry breaking scale, Φ 16 is still living in the SM thermal bath by interaction with SM fermions induced by virtual SM Higgs. By comparing the relevant interaction rate of the diagram in Fig. 8 to Hubble expansion rate it is realized that Φ 16 would easily pair-annihilate to SM fermions at T SM m 16 . Here m f is a SM fermion mass and m h is the physical Higgs particle mass. and photon density is ρ γ (a BBN ) = π 2 30 × 2 × (1MeV) 4 (66)

D Mapping the thermal WDM mass to a non-thermal WDM
It was observed in Ref. [59] that the linear matter power spectra associated with different WDM models are very similar when the same variance of velocity and the comoving Jean scale (k J ) are assumed. The comoving Jean scale at the matter-radiation equality time is defined as [59] k J = a 4πGρ m σ 2 a=aeq , where ρ m is the matter density and σ is velocity variance of DM.
In accordance with this, it was argued in Ref. [65] that equating the warmness parameters for the thermal WDM and WDM of another type differing from the thermal one constructs the map between masses. The warmness parameter (σ ≡σT /m) of a WDM introduced in [65] is defined with temperature T , mass m and the quantitỹ where f (p) is the momentum space distribution function and q ≡ p/m is used. To establish the map from the thermal WDM mass to another WDM candidate (χ), one can begin with where σ χ is the warmness of χ-WDM and σ wdm is that of the early decoupled thermal WDM. This equation tells us that once one knows T χ , T wdm andσ χ at a = a eq , one can map the constraint on m wdm to that on m χ , knowingσ wdm =3.6 from Fermi-Dirac distribution. T χ andσ χ are closely related to production mechanism of χ-WDM. On the other hand, for the early decoupled thermal WDM, T wdm is determined by DM relic density. Today, comparison of thermal WDM to the neutrino gives [66] Ω wdm h 2 0.12 = m wdm 94eV where T ν,0 = (4/11) 1/3 T γ,0 is today's neutrino temperature.

E Would-be temperature of DM candidate
The necessary condition that fermion DM candidate should satisfy to form a cored halo profile within a dSphs is that its "would-be" temperature today (T DM,0 ) in the absence of structure formation should be smaller than a degeneracy temperature for the dSphs (T DEG ) [28]. From the property that DM's momentum scales as ∼ a −1 and the temperature of DM can be defined via E k ∼ kT , we can infer that DM's temperature scales as ∼ a −1 for relativistic state and ∼ a −2 for non-relativistic state.
For the case where fermion DM candidate is produced from a non-relativistic scalar decay and free-stream since then, the scale factor (a NR ) at which DM becomes nonrelativistic is given by where m S is the mother scalar's mass and a FS is the scale factor at which DM starts free-streaming. Therefore, starting with p DM (a FS ) m S /2, the "would-be" temperature for DM today is computed bỹ where Eq. (71) is used for the second equality. The degeneracy temperature for a dSphs used for checking is roughly T DEG O(10 −4 )K − O(10 −3 )K [28].