Non-linear growth (e.g., N-body simulation) 3124 A. Schneider lines) are all plotted in Fig. 4. For the case of CDM (black), the sharp-k mass function closely follows both the simulation measure- ments and the Sheth–Tormen model. For the case of WDM (red, cyan, purple, pink), MDM (green, magenta, blue), and WIMP DM (brown, orange), the sharp-k mass function gives a reasonably good match to simulations, while the Sheth–Tormen approach fails to match the ﬂattening or the turnaround visible in simulations. In Schneider et al. (2013), the sharp-k model has been reported to underestimate the halo abundance when the suppression scale lies in the exponential tail of the halo mass function (i.e. for ν 1), which generally happens at very high redshift. It turns out, however, that this discrepancy between the sharp-k model and the data is greatly reduced for haloes deﬁned by a spherical overdensity instead of a friends-of-friends linking criterion (see Watson et al. 2013, for a comparison of the two). We therefore do not use the correction model proposed by Schneider et al. (2013). 4.3 Conditional mass function Another important application of the EPS model is the conditional Downloaded from https://academic.oup • ఱͷۜՏͷ؍ଌ͞ΕͨαςϥΠτۜՏͷ: • ͍ܰ҉ࠇ࣭free-streaming ʹΑΓখنͳߏΛݮΒͯ͠͠·͏ ɹˠ Λ༧ݴ͢ΔΑ͏ͳཧغ٫ Nsat ≃ 63 Nsat ≪ 63 [Murgia et al. (2017)] [Schneider (2015)] ఱͷۜՏʹ͍ΔΑΓখ͍ۜ͞Տ
Linear perturbation ؍ଌྔ (ྫ:αςϥΠτۜՏͷ) Non-linear growth (e.g., N-body simulation) • DMͷˠϘϧπϚϯํఔࣜͷੵ J 2-body s-ch t-ch 3-body Fermi-Dirac () () fDM (tf , p) = ∫ tf ti dt 1 EDM C t, a (tf) a(t) p gDM EDM C (t, pDM) = 1 2EDM ∫ ∏ i≠DM d3pi (2π)32Ei (2π)4δ4 (Pi − Pf) × ∑ spin ℳ 2 f1 f2 ⋯(1 ∓ f3) (1 ∓ f4)⋯ 4. Einstein Equations and Energy-Momentum Conservation For a homogeneous Friedmann-Robertson-Walker universe with energy density ¯ ρ(τ) and pressure ¯ P(τ), the Einstein equations give the following evolution equations for the expansion factor a(τ): ˙ a a 2 = 8π 3 Ga2 ¯ ρ − κ , (19) d dτ ˙ a a = − 4π 3 Ga2(¯ ρ + 3 ¯ P) , (20) where the dots denote derivatives with respect to τ, and κ is positive, zero, or negative for a closed, ﬂat, or open universe, respectively. We consider only models with total Ω = 1 in this paper, so we set κ = 0. A cosmological constant is allowed through its inclusion in ¯ ρ and ¯ P: ¯ ρΛ = Λ/8πG = − ¯ PΛ. This is the only place that Λ enters in the entire set of calculations. It follows from equation (19) with κ = 0 that the expansion factor scales as a ∝ τ in the radiation-dominated era, a ∝ τ2 in the matter-dominated era, and a ∝ (τ∞ −τ)−1 in a cosmological constant-dominated era (in the latter case, τ∞ is the radius of the de Sitter event horizon). We ﬁnd it most convenient to solve the linearized Einstein equations in the two gauges in the Fourier space k. In the synchronous gauge, the scalar perturbations are characterized by h(k, τ) and η(k, τ) in equation (4). In terms of h and η, the time-time, longitudinal time-space, trace space-space, and longitudinal traceless space-space parts of the Einstein equations give the following four equations to linear order in k-space: Synchronous gauge — k2η − 1 2 ˙ a a ˙ h = 4πGa2δT0 0(Syn) , (21a) k2 ˙ η = 4πGa2(¯ ρ + ¯ P)θ(Syn) , (21b) ¨ h + 2 ˙ a a ˙ h − 2k2η = −8πGa2δTi i(Syn) , (21c) ¨ h + 6¨ η + 2 ˙ a a ˙ h + 6 ˙ η − 2k2η = −24πGa2(¯ ρ + ¯ P)σ(Syn) . (21d) The label “Syn” is used to distinguish the components of the energy-momentum tensor in the synchronous gauge from those in the conformal Newtonian gauge. The variables θ and σ are deﬁned as (¯ ρ + ¯ P)θ ≡ ikjδT0 j , (¯ ρ + ¯ P)σ ≡ −(ˆ ki ˆ kj − 1 3 δij)Σi j , (22) and Σi j ≡ Ti j − δi jTk k/3 denotes the traceless component of Ti j. Kodama & Sasaki (1984) deﬁne the anisotropic stress perturbation Π, related to our σ by σ = 2Π ¯ P/3(¯ ρ + ¯ P). When the diﬀerent • ΞΠϯγϡλΠϯํఔࣜΛઢܗۙࣅͰղ͘ → Linear matter power spectrum CDM 2.0 keV 5.3 keV - - - - () • N-body simulation ͨ͘͞Μͷؒͱܭࢿݯ͕ඞཁ
Schneider lines) are all plotted in Fig. 4. For the case of CDM (black), the sharp-k mass function closely follows both the simulation measure- ments and the Sheth–Tormen model. For the case of WDM (red, cyan, purple, pink), MDM (green, magenta, blue), and WIMP DM (brown, orange), the sharp-k mass function gives a reasonably good match to simulations, while the Sheth–Tormen approach fails to match the ﬂattening or the turnaround visible in simulations. In Schneider et al. (2013), the sharp-k model has been reported to underestimate the halo abundance when the suppression scale lies in the exponential tail of the halo mass function (i.e. for ν 1), which generally happens at very high redshift. It turns out, however, that this discrepancy between the sharp-k model and the data is greatly reduced for haloes deﬁned by a spherical overdensity instead of a friends-of-friends linking criterion (see Watson et al. 2013, for a comparison of the two). We therefore do not use the correction model proposed by Schneider et al. (2013). 4.3 Conditional mass function Another important application of the EPS model is the conditional mass function, which gives the abundance of haloes per mass and look-back redshift z1 , eventually ending up in a single host halo at redshift z0 . As the conditional mass function provides a connection between haloes at different redshifts, it acts as the starting point of more evolved quantities such as the halo collapse redshift, the number of satellites, and halo merger trees. The conditional mass function is given by dN(M|M0 ) d ln M = − M0 M Sf (δc, S|δc,0, S0 ) d ln S d ln M (13) (Lacey & Cole 1993). For the sharp-k model this can be simpliﬁed to dNSK (M|M0 ) d ln M = 1 6π2 M0 M f (δc, S|δc,0, S0 ) P(1/R) R3 , (14) where the ﬁlter scale R and the mass M are related by equation (12). The conditional ﬁrst-crossing distribution again depends on the assumed model for non-linear collapse. The case of spherical col- lapse is given by f (δc, S|δc,0, S0 ) = δc − δc,0 √ 2π(S − S0 ) exp − (δc − δc,0 )2 2(S − S0 ) , (15) Figure 5. Conditional mass functions for a M0 = 1013 h−1 M host and a look-back redshift of z = 1.1. Coloured symbols refer to simulation outputs (with circumjacent shaded regions representing the uncertainty due to arte- fact subtraction), while the solid and dotted lines represent the sharp-k model and the standard Press–Schechter model, respectively. The colour-coding is the same as in the previous plots. 4.4 Estimating the number of dwarf satellites Each DM scenario has to produce a sufﬁcient amount of substruc- tures to account for the observed MW satellites. While some (or most) of the substructures could be dark due to inefﬁcient star for- mation, fewer substructures than observed means the failure of the DM scenario. In the case of WDM, comparing numbers of simulated substructures with observed satellites has led to tight constraints on the thermal particle mass ruling out masses below 2 keV (Polisen- sky & Ricotti 2011; Kennedy et al. 2014). The EPS approach can be used to estimate the average number of dwarf galaxies orbiting a galaxy like the MW. This means it is possible to check whether a certain DM scenario is likely to be in agreement with observations without running expensive numerical zoom-simulations of an MW halo. In principle, ﬁnding the number CDM 2.0 keV 5.3 keV - - - - [/] () Particle model ℒSM + ℒDM Linear perturbation ؍ଌྔ (ྫ:αςϥΠτۜՏͷ) Non-linear growth (e.g., N-body simulation) P(k): linear matter power spectrum ֤εέʔϧͰͷߏͷଟ͞Λද͢ྔ
• Cons: • NNͷ܇࿅ͷͨΊʹɺ্ͷҰ࿈ͷܭΛҰճΔඞཁ͕͋Δʢैདྷͷख๏͕ෆཁʹͳΔΘ͚Ͱͳ͍ʣ NNͰϑΟοςΟϯάؔΛ༩͑Α͏ Nsat = f(P(k)) P(k) = f(ℒDM ) 3124 A. Schneider lines) are all plotted in Fig. 4. For the case of CDM (black), the sharp-k mass function closely follows both the simulation measure- ments and the Sheth–Tormen model. For the case of WDM (red, cyan, purple, pink), MDM (green, magenta, blue), and WIMP DM (brown, orange), the sharp-k mass function gives a reasonably good match to simulations, while the Sheth–Tormen approach fails to match the ﬂattening or the turnaround visible in simulations. In Schneider et al. (2013), the sharp-k model has been reported to underestimate the halo abundance when the suppression scale lies in the exponential tail of the halo mass function (i.e. for ν 1), which generally happens at very high redshift. It turns out, however, that this discrepancy between the sharp-k model and the data is greatly reduced for haloes deﬁned by a spherical overdensity instead of a friends-of-friends linking criterion (see Watson et al. 2013, for a comparison of the two). We therefore do not use the correction model proposed by Schneider et al. (2013). 4.3 Conditional mass function Another important application of the EPS model is the conditional mass function, which gives the abundance of haloes per mass and look-back redshift z1 , eventually ending up in a single host halo at redshift z0 . As the conditional mass function provides a connection between haloes at different redshifts, it acts as the starting point of more evolved quantities such as the halo collapse redshift, the number of satellites, and halo merger trees. The conditional mass function is given by dN(M|M0 ) d ln M = − M0 M Sf (δc, S|δc,0, S0 ) d ln S d ln M (13) (Lacey & Cole 1993). For the sharp-k model this can be simpliﬁed to dNSK (M|M0 ) d ln M = 1 6π2 M0 M f (δc, S|δc,0, S0 ) P(1/R) R3 , (14) where the ﬁlter scale R and the mass M are related by equation (12). The conditional ﬁrst-crossing distribution again depends on the assumed model for non-linear collapse. The case of spherical col- lapse is given by f (δc, S|δc,0, S0 ) = δc − δc,0 √ 2π(S − S0 ) exp − (δc − δc,0 )2 2(S − S0 ) , (15) Figure 5. Conditional mass functions for a M0 = 1013 h−1 M host and a look-back redshift of z = 1.1. Coloured symbols refer to simulation outputs (with circumjacent shaded regions representing the uncertainty due to arte- fact subtraction), while the solid and dotted lines represent the sharp-k model and the standard Press–Schechter model, respectively. The colour-coding is the same as in the previous plots. 4.4 Estimating the number of dwarf satellites Each DM scenario has to produce a sufﬁcient amount of substruc- tures to account for the observed MW satellites. While some (or most) of the substructures could be dark due to inefﬁcient star for- mation, fewer substructures than observed means the failure of the DM scenario. In the case of WDM, comparing numbers of simulated substructures with observed satellites has led to tight constraints on the thermal particle mass ruling out masses below 2 keV (Polisen- sky & Ricotti 2011; Kennedy et al. 2014). The EPS approach can be used to estimate the average number of dwarf galaxies orbiting a galaxy like the MW. This means it is possible to check whether a certain DM scenario is likely to be in agreement with observations without running expensive numerical zoom-simulations of an MW halo. In principle, ﬁnding the number CDM 2.0 keV 5.3 keV - - - - [/] () Particle model ℒSM + ℒDM Linear perturbation ؍ଌྔ (ྫ:αςϥΠτۜՏͷ) Non-linear growth (e.g., N-body simulation) P(k): linear matter power spectrum ֤εέʔϧͰͷߏͷଟ͞Λද͢ྔ
model We consider Majorana fermion DM , which feebly interacts with a Dirac fermion and a complex scalar . also couples to a Dirac fermion f. We discuss the correspondence to prominent FIMP models in section 3.3. The Lagrangian relevant to freeze-in production of is LF.I. = y ¯ yf ¯ ff + h.c. , (2.1) where y and yf are Yukawa couplings. We assume that and f are in equilibrium with the thermal plasma. In the following, we consider the mass spectrum of m , mf ⌧ m < m .4 Due to the feeble interaction with the other particles, y ⌧ 1, is not in equilibrium with the thermal plasma. Meanwhile is produced by freeze-in processes: • 2-body decay: ! , ¯ ! ⇤ • t-channel scattering: f ! f, ¯ f ! ¯ f, ¯ f ! f, ¯ ¯ f ! ¯ f • s-channel scattering: f ¯ f ! , f ¯ f ! ¯ The scattering processes are mediated by . The freeze-in production is most e cient when the heaviest particle in the process, i.e., , becomes non-relativistic. After that the pro- duction is suppressed by the Boltzmann factor. Thus we deﬁne the decoupling temperature Tdec = m . Model Parameters (mχ , mΨ /mϕ , yf , Δ) T2(k) = P(k) PCDM (k) ∼ (1 + (αk)β)2γ Nsat DM Transfer function ΛύϥϝτϥΠζ αςϥΠτۜՏͷ (α, β, γ) ରԠ͢Δ۩ମతͳϞσϧͷྫ • εςϥΠϧɾχϡʔτϦϊ • Axino in DFSZ SUSY model χ = χ = Figure 4: Schematic diagram of the neural network. Partly taken from Ref. [196]. Figure 4: Schematic diagram of the neural network. Partly taken from Ref. [196]. ਖ਼نͷखଓ͖Ͱ܇࿅σʔλΛ࡞Δ → Hidden layer ೋͭͷ (shallow) neural network Λ܇࿅