}\Lambda_{\rm TC}$ the extra TCb $NNN^*$ (denoted with a $^*$ and not included in the list above because it has spin 3/2) could become the stable DM candidate; at the same time the $\SU(4)_{\rm TF}$ classification breaks down. \paragraph{\boldmath c) SU($N$)$_{\rm TC}$ models $\Q=N \oplus L$ and $\Q=N \oplus L \oplus \tilde E$.} In both models, enough Yukawa couplings are allowed such that only techni-baryon number is conserved and all TC$\pi$ are unstable. For the $N \oplus L$ ($\NF=3$) and the $N \oplus L \oplus \tilde E$ ($\NF=4$) models respectively, these are: \beq\label{eq:TCpiNL} \begin{array}{l} \hbox{TC$\pi$ : } 8 = 1_0 \oplus 2_{\pm 1/2} \oplus 3_0 \\ \hbox{TC$\pi$ : } 15 = 1_{2 \times 0,\pm 1} \oplus 2 \times 2_{\pm 1/2} \oplus 3_0 \end{array}\qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$}. \eeq For $N=3$, the spin 1/2 TCb do not contain any DM candidate, for example in the the $N \oplus L$ model they are \beq \label{N+L-TCb} \hbox{TCb : } 8 = 1_{-1} \oplus 2_{-1/2,-3/2} \oplus 3_{-1} \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$}. \eeq The DM candidate is the singlet $NNN^*$, which only exists with spin 3/2. Thereby these models are viable only as long as the techni-quark masses $m_L$ and $m_{\tilde E}$ are of order $\Lambda_{\rm TC}$ and large enough that $NNN^*$ is the lightest TCb. This state lies in the 10 of $\SU(3)_{\rm TF}$ in the $N\oplus L$ model \beq \hbox{TCb* : } 10 = 1_0 \oplus 2_{-1/2} \oplus 3_{-1} \oplus 4_{-3/2} \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$} \eeq and in the $\overline{20}''$ of $\SU(4)_{\rm TF}$ in the $N\oplus L \oplus \tilde E$ model. The same is true for $N=4$, where the only DM candidate is the singlet $NNNN^*$ that lies in the completely symmetric spin 2 representation $\yng(4)$\ . In the $N \oplus L$ model, this representation decomposes as \begin{eqnarray} \hbox{TCb$^*$ : } 15' &=& 1_0 \oplus 2_{-1/2} \oplus 3_{-1} \oplus 4_{-3/2} \oplus 5_{-2} \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$}. \end{eqnarray} The $N \oplus L$ model is allowed by perturbativity constraints up to $N=14$, while the $N \oplus L \oplus \tilde E$ is allowed up to $N=5$ (with increasing spin of the DM candidate). \paragraph{\boldmath d) SU($N$)$_{\rm TC}$ models $\Q = V\oplus L$ and $V \oplus L \oplus \tilde E$.} Other possible extensions of the first model are $\Q = V\oplus L$ and $V \oplus L \oplus \tilde E$. A possible problem of these models is that, even for $N=3$, the $\SU(2)_L$ gauge coupling becomes non perturbative around $10^{17}\GeV$. In view of the Yukawa couplings $V L H$, $\tilde E L H$, all TC$\pi$ are unstable and given by \beq \hbox{TC$\pi$ : } 24 = 1_0 \oplus 2_{\pm 1/2} \oplus 2 \times 3_0 \oplus 4_{\pm 1/2} \oplus 5_0 \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$} \,, \label{LVpions} \eeq in the $V \oplus L$ model, and by \beq \hbox{TC$\pi$ : } 35 = 2 \times 1_0 \oplus 2 \times 2_{\pm 1/2} \oplus 3_{2 \times 0,\pm 1} \oplus 4_{\pm 1/2} \oplus 5_0 \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$} \,, \eeq in the $V \oplus L \oplus \tilde E$ model. In both models the TCb DM candidate is the $VVV$ state that forms a weak triplet as in the $\Q = V$ model: the extra techni-quarks $L$ (and possibly $\tilde E$) does not lead to any extra DM candidates and play a minor role provided that they are heavy enough. In the $V \oplus L$ model, the lightest TCb multiplet is a $\overline{40}$ of $\SU(5)_{\rm TF}$ that decomposes as: \beq \hbox{TCb : } \overline{40} = 1_{-1} \oplus 2_{2 \times (-1/2),-3/2} \oplus 3_{0, 2 \times (-1)} \oplus 2 \times 4_{-1/2} \oplus 5_{0,-1} \oplus 6_{-1/2}\eeq under $\SU(2)_L\otimes{\rm U}(1)_Y$. \paragraph{\boldmath e) SU($N$)$_{\rm TC}$ models $\Q = V\oplus N\oplus L$ and $\Q = V \oplus N \oplus L \oplus \tilde E$.} As in the previous models, sub-Planckian Landau poles are avoided only for $N=3$ (where $g_2$ becomes non perturbative around $10^{17} \GeV$). Since $L$ and $\tilde E$ cannot enter in an hypercharge-less TCb, the DM candidates are the same of the $V \oplus N$ model. Unlike in the $V\oplus N$ model, the Yukawa couplings $V L H$, $N L H$ and $L \tilde E H$ break all species number symmetries, such that all TC$\pi$ are unstable. In the $V \oplus N \oplus L$ model ($\NF=6$), the TC$\pi$ are \beq \hbox{TC$\pi$ : } 35 = 2 \times 1_0 \oplus 2 \times 2_{\pm 1/2} \oplus 4 \times 3_0 \oplus 4_{\pm 1/2} \oplus 5_0 \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$} \,. \eeq In the $V \oplus N \oplus L \oplus \tilde E$ model ($\NF=7$), the list extends to \beq \hbox{TC$\pi$ : } 48 = 1_{3 \times 0,\pm 1} \oplus 3 \times 2_{\pm 1/2} \oplus 3_{4 \times 0,\pm 1} \oplus 4_{\pm 1/2} \oplus 5_0 \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$} \,. \eeq \paragraph{\boldmath f) SU($N$)$_{\rm TC}$ model $\Q=N \oplus L \oplus \tilde L$.} The model allows two Yukawa couplings ($N L H$, $N \tilde L H$) such that there are no stable TC$\pi$ and allows for DM TCb candidates not present in the previous models. The unstable TC$\pi$ are: \beq \label{eq:TCpiNLL} \hbox{TC$\pi$ : } 24 = 1_{2 \times 0,\pm 1} \oplus 2\times 2_{\pm 1/2} \oplus 3_{2 \times 0,\pm 1} \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$} \,. \eeq Sub-Planckian Landau poles are avoided for $N\le 7$. Here we discuss the TCb DM candidates for $N=3,4$. For $N=3$, the lighter TCb fill a $\overline{40}$ of $\SU(5)_{\rm TF}$ that decomposes as \beq \label{N+L+tL-TCb} \hbox{TCb : } \overline{40} = 1_{2 \times 0,\pm 1} \oplus 2_{3 \times (\pm 1/2),\pm 3/2} \oplus 3_{2 \times 0,\pm 1} \oplus 4_{\pm 1/2} \eeq \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$}, so that the TCb DM candidates are singlets made of $N \tilde L L$. For $N=4$, the lighter TCb are \beq \label{TCbNLL} \hbox{TCb : } \overline{50} = 1_{3 \times 0,\pm 1,\pm 2} \oplus 2_{2 \times (\pm 1/2),\pm 3/2} \oplus 3_{2 \times 0, 2 \times (\pm 1)} \oplus 4_{\pm 1/2} \oplus 5_0 \,,\eeq under $\SU(2)_L\otimes{\rm U}(1)_Y$. The TCb DM candidates are singlets made of $LL \tilde L \tilde L$ and $L \tilde L NN$. \paragraph{\boldmath g) SU($N$)$_{\rm TC}$ model $\Q = N\oplus L \oplus \tilde L \oplus \tilde E$.} This is a non trivial extension of the previous model, with one more Yukawa coupling allowed ($L \tilde E H$), so that there are no stable TC$\pi$. The model is allowed only for $N=3,4$, since for greater values of $N$ the coupling $g_Y$ develops a sub-Planckian Landau pole. The unstable TC$\pi$ can be listed as: \beq \label{eq:TCpiNLLE} \hbox{TC$\pi$ : } 35 = 1_{3 \times 0, 2 \times (\pm 1)} \oplus 2_{3 \times (\pm 1/2),\pm 3/2} \oplus 3_{2 \times 0,\pm 1} \qquad \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$} \,. \eeq This model gives a TCb DM candidate not present in the previous models: $\tilde L \tilde L \tilde E$ and $N \tilde L \tilde L \tilde E$ for $N=3$ and $N=4$ respectively. For $N=3$, the lightest multiplet of TCb decomposes under $\SU(2)_L\otimes{\rm U}(1)_Y$ as \begin{eqnarray} \hbox{TCb : } 70 &=& \hbox{TCb}_{N\oplus L \oplus \tilde L} \oplus 1_{0, 3 \times (-1), 2 \times (-2)} \oplus 2_{2 \times (-1/2), 3 \times (-3/2),(-5/2)} \oplus 3_{0, 2 \times (-1),-2} \,, \end{eqnarray} where $\hbox{TCb}_{N\oplus L \oplus \tilde L}$ is defined in eq.~\eqref{N+L+tL-TCb}. For $N=4$ we get \begin{eqnarray} \hbox{TCb : } \overline{105'} &=& \hbox{TCb}_{N \oplus L \oplus \tilde L} \oplus 1_{0,2 \times (-1),3 \times (-2)} \oplus 2_{1/2, 3 \times (-1/2), 4 \times (-3/2), 2 \times (-5/2)} \notag \\ &&\oplus 3_{0, 3 \times (-1), 2 \times (-2),-3} \oplus 4_{-1/2,-3/2} \end{eqnarray} where now $\hbox{TCb}_{N \oplus L \oplus \tilde L}$ refers to eq.~\eqref{TCbNLL}. In each case, besides the TCb DM candidates of the $N \oplus L \oplus \tilde L$ model, there a singlet DM candidate made of $\tilde L \tilde L \tilde E$ or $N \tilde L \tilde L \tilde E$. \paragraph{\boldmath h) SU($N$)$_{\rm TC}$ model $\Q = L \oplus \tilde L \oplus E \oplus \tilde E \oplus N$.} The model has $\NF=7$ and for $N=3$ gives $\Delta b_Y=12$ so that hypercharge has a Landau pole around the Planck scale, so that it cannot be extended to $N>3$. Thanks to the presence of $N$, it allows for 4 Yukawa couplings ($L \tilde E H$, $\tilde L E H$, $LNH$, $\tilde{L}NH$) that break all species number symmetries. The unstable TC$\pi$ are: \beq \hbox{TC$\pi$ : } 48 = 1_{4 \times 0, 3 \times (\pm 1),\pm 2} \oplus 2_{4 \times (\pm 1/2), 2 \times (\pm 3/2)} \oplus 3_{2 \times 0,\pm 1} \eeq \hbox{under $\SU(2)_L\otimes{\rm U}(1)_Y$}. The lightest TCb fill a $112$ of $\SU(7)_{\rm TF}$, that decomposes as \begin{eqnarray} \hbox{TCb : } 112 &=& \hbox{TCb}_{N\oplus L \oplus \tilde L} \oplus 1_{4 \times (0, \pm 1), 2 \times (\pm 2)} \oplus 2_{4 \times (\pm 1/2), 3 \times(\pm 3/2), \pm 5/2} \oplus 3_{2 \times (0,\pm 1),\pm 2} \,, \end{eqnarray} under $\SU(2)_L \otimes {\rm U}(1)_Y$. The TCb DM candidates are those of the $N \oplus L \oplus \tilde L$ model, defined in eq.~\eqref{N+L+tL-TCb}, plus the singlets $LLE$, $\tilde L \tilde L \tilde E$ and $E \tilde E N$. We next consider models with coloured techni-quarks. \paragraph{\boldmath i) SU($N$)$_{\rm TC}$ model $\Q = Q\oplus \tilde D$.} The simplest golden-class model with colored techni-quarks is $\Q=Q\oplus \tilde D$, that is allowed for $N=3,4$ and gives a DM candidate only for $N=3$. The model has $\NF=6$ and does not lead to unwanted stable states because species number is broken by the Yukawa coupling $Q\tilde{D} H$. The model predicts a set of unstable TC$\pi$ in the $80$ representation of $\SU(9)_{\rm TF}$, that decomposes under the SM gauge group $\SU(3)_c \otimes \SU(2)_L \otimes {\rm U}(1)_Y$ as \beq \hbox{TC$\pi$ : } 80= (1,1)_0 \oplus (1,2)_{\pm 1/2} \oplus (1,3)_0 \oplus 2(8,1)_0 \oplus (8,2)_{\pm 1/2} \oplus (8,3)_0.\eeq For $N=3$ the multiplet of lighter TCb fills a $240$ of $\SU(9)_{\rm TF}$, that decomposes as \begin{eqnarray} \hbox{TCb : } 240 &=& \nonumber (1,1)_0 \oplus (1,2)_{\pm 1/2} \oplus (1,3)_0 \oplus (8,1)_{2 \times 0,-1} \oplus (10,1)_0 \oplus (8,2)_{1/2, 2 \times (-1/2)} \notag \\ &&\oplus (10,2)_{\pm 1/2} \oplus 2(8,3)_0 \oplus (10,3)_0 \oplus (8,4)_{1/2} \end{eqnarray} under the SM gauge group $\SU(3)_c \otimes \SU(2)_L \otimes {\rm U}(1)_Y$. The DM candidate is the neutral singlet $QQ\tilde D$, which can be the lightest TCb. \paragraph{\boldmath l) SU($N$)$_{\rm TC}$ model $\Q = Q\oplus \tilde D\oplus \tilde U$.} This extension of the previous model allows for two Yukawa couplings, $Q H \tilde D$ and $Q H \tilde U$, so that there are no stable TC$\pi$. This model has $\NF=12$ and is allowed only for $N=3$, where $\Delta b_3 = 8$. It predicts an extended set of unstable TC$\pi$, that fills a $143$ of $\SU(12)_{\rm TF}$: \beq \hbox{TC$\pi$ : } 143= \hbox{TC$\pi$}_{Q \oplus \tilde D} \oplus (1,1)_{0,\pm 1} \oplus (1,2)_{\pm 1/2} \oplus (8,1)_{0,\pm 1} \oplus (8,2)_{\pm 1/2} \eeq under $\SU(3)_c \otimes \SU(2)_L \otimes {\rm U}(1)_Y$. The model contains two TCb DM candidates: $QQ \tilde D$ and $\tilde D \tilde D \tilde U$. The lighter TCb fill a $572$ of $\SU(12)_{TF}$, that decomposes as \begin{eqnarray} \hbox{TCb : } 572 &=& \hbox{TCb}_{Q \oplus \tilde D} \oplus (1,1)_{0,2 \times 1} \oplus (1,2)_{2 \times 1/2,3/2} \oplus (1,3)_1 \oplus (8,1)_{2 \times 0, 4 \times 1,2} \!\oplus\! (10,1)_{0, 2 \times 1} \notag \\ && \oplus (8,2)_{4 \times 1/2, 2 \times 3/2} \oplus (10,2)_{2 \times 1/2,3/2} \oplus 2 \times (8,3)_1 \oplus (10,3)_1 \end{eqnarray} under $\SU(3)_c \otimes \SU(2)_L \otimes {\rm U}(1)_Y$. The TCb DM candidates are those of the $\Q = Q\oplus \tilde D$ model plus a singlet made of $\tilde D \tilde D \tilde U$. Notice that colored techni-quarks never provide golden-class models for $N\ge 4$. For example the model $\Q=G$ leads, for $N=4$, to an acceptable TCb DM candidate; but $g_3$ develops a sub-Planckian Landau pole.\footnote{Detailed group-theoretical computations show that the simplest model $\Q=G$ leads, for $N=3$ only to coloured lighter TCb; this can be cured by adding extra techni-quarks (e.g.\ $\Q = G\oplus N$) but their addition lead to stable coloured TC$\pi$ or TCb. Furthermore $g_3$ develops a Landau pole below the Planck scale. So these are not golden-class models. } Landau poles also exclude the model $\Q = Q\oplus \tilde U \oplus \tilde D$ (two Yukawa couplings allowed, no stable TC$\pi$) that for $N=6$ provides a TCb DM candidate, $QQ\tilde D\tilde D\tilde D\tilde U$. ]]>

v_{\rm min}=\sqrt{M_{\cal N} E_R/2\mu^2}$, which could be tested relying on the seasonal variation in the average $v^2$. We parameterize the dipole moments in terms of their gyro-magnetic and gyro-electric constant $g_M$ and $g_E$ as in eq.\eq{gfactors}. Composite DM generically predicts an order one gyromagnetic factor $g_M$, and a possibly sizeable gyro-electric factor $g_E \sim \theta_{\rm TC} {\rm Min}[m_{\Q}]/M_{\rm DM}$ as discussed in section~\ref{spec}. This means that for $M_{\rm DM}\approx 100\TeV$ and $g_M\sim 1$ the magnetic effect is 3 orders of magnitude below the experimental limit, \beq \sigma_{\rm SI}<10^{-44}\cm^2 \frac{M_{\rm DM}}{\TeV}\qquad \hbox{for $M_{\rm DM}\gg M_{\cal N}$}\eeq and at the level of the neutrino background, see also~\cite{Appelquist-ml-2013ms}. The electric effect is comparable to the present LUX bound for $g_E \approx 0.01$ and $M_{\rm DM}\approx 100\TeV$, as illustrated in figure~\ref{fig:sigmaDD}a. In some models DM has chromo-dipoles, that lead to a similar scattering rate with $e^4 Z^2/E_R$ replaced by $g_3^4 /\Lambda_{\rm QCD}$ times a nuclear form factor, which is strongly suppressed at energies below $\Lambda_{\rm QCD}$. Thereby chromo-dipoles do not compete with electric dipoles. Some composite DM models predict that DM is a TCb with higher-spin. Spin 1 DM can have characteristic spin-dependent interactions which are, however, suppressed by the transferred momentum~\cite{Krauss}. More interestingly, a composite spin-1 TCb $B_\mu$ could have a dimension-4 interaction $B_\mu B^*_\nu F^{\mu\nu}$ with a photon. Even when the lighter TCb is mostly composed of neutral SM singlets $N$, it also contains a small component of charged heavier techni-quarks with a momentum asymmetry (an effect analogous to the strange momentum asymmetry in nucleons~\cite{NuTeV}). \begin{figure} \centerline{\includegraphics[width=0.45\textwidth]{figs/CompositeDM-SI}\qquad \includegraphics[width=0.45\textwidth]{figs/CompositeDM-SD}} ]]>

}100\,{\rm keV}$. A smaller mass difference can be obtained for $y \sim 10^{-3}$ and gives rise to inelastic DM phenomenology~\cite{TuckerSmith-ml-2001hy}. ]]>

\Lambda_{\rm TC}$. In the more interesting regime $m_{L,V}<\Lambda_{\rm TC}$ techni-color effects cannot be neglected and the loops will be dominated by the hadrons of the theory, as depicted in the right-handed panel of figure~\ref{fig:edm2loops}. A detailed study will appear in~\cite{ACDMcollider}. ]]>

3$~\cite{Pis}. The phase transition occurs, within our framework, at a temperature $T\sim \Lambda_{\rm TC}$ (in the thermal dark matter scenario $ \Lambda_{\rm TC}\sim \mathcal{O}(10 {\rm \ TeV})$) and can lead to large anisotropic fluctuations in the energy momentum tensor sourcing the gravitational waves (GW) in the early universe. Following~\cite{Schwaller-ml-2015tja}, we estimate the frequency of the peak in the GW signal as a function of the phase transition temperature $T$ as: \beq f_{\rm peak} = 3.3 \times 10^{-3}\ {\rm Hz} \times \Big(\frac{T}{10\, {\rm TeV}}\Big) \times \Big( \frac{\beta}{10H}\Big) \eeq where $\beta$ is the duration of the phase transition which is usually taken in the range 1-100 of a Hubble time $H$. For the reference values of the parameters, the amplitude of the expected GW signal is $h^2 \Omega_{\rm GW}\sim 10^{-9}$ ~\cite{Schwaller-ml-2015tja} which is in the range that can be probed by future satellite experiments such as (E)LISA~\cite{ELISA}. \begin{figure} \centerline{ \includegraphics[width=0.45\textwidth]{figs/GUT-lattice.pdf}\qquad \includegraphics[width=0.45\textwidth]{figs/GUT-QD.pdf} } ]]>