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+{"text":"\\section*{Acknowledgments}\n\nWe thank A. Vrajitoarea, P. Groszkowski, N. Earnest and A. Shearrow for helpful discussions. Work at Princeton, Northwestern and Chicago was supported by Army Research Office Grant No. W911NF-1910016. Devices were fabricated in the Princeton University Quantum\nDevice Nanofabrication Laboratory and in the Princeton Institute for the Science and Technology of Materials (PRISM) cleanroom. The authors acknowledge the use of Princeton's Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation (NSF)-MRSEC program (DMR-1420541). This work was undertaken in part thanks to funding from NSERC and the Canada First Research Excellence Fund.\n\n  \n\\section*{Supplementary materials}\nMaterials and Methods\\\\\nSupplementary Text\\\\\nFigs.~S1 to S7\\\\\nTables S1 to S4\\\\\nReferences \\cite{johansson,kohn}\n\n\n\\clearpage\n\n\\begin{figure}[htp]\n    \\centering\n    \\includegraphics[width=17cm]{figure1_v5.pdf}\n\\end{figure}\n\\begin{spacing}{1.0}\n\\noindent {\\bf Fig.~1.} \\textbf{The $\\mathbf{0-\\pi}$ superconducting qubit and its energy level structure.} (\\textbf{A}) The circuit diagram of the $0-\\pi$ qubit \\cite{kitaev1,brooks}. The circuit has one closed loop with four nodes connected by a pair of Josephson junctions ($E_J, C_J$), large capacitors ($C$) and superinductors ($L$). (\\textbf{B}) Left panel: the $V(\\theta,\\phi)$ double-well potential landscape of the circuit in the absence of magnetic fields. The ground state of the 0 valley is localized along $\\theta=0$ (middle panel), while the lowest-lying state of the $\\pi$ valley along $\\theta=\\pi$ (right panel). The line cuts along the two valleys in the $\\phi$ direction show that the potential resembles a fluxonium potential. (\\textbf{C}) The four modes of the $0-\\pi$ circuit with colors of the nodes indicating the sign of normal-mode amplitudes. (\\textbf{D}) Left panel: schematic of the symmetric and antisymmetric ground states of the $\\pi$ valley. The hybridization of these states leads to a magnetic sweet spot (right panel). (\\textbf{E}) The two-dimensional wavefunctions of the eigenstates, which are located mostly in the 0 (left) or in the $\\pi$ (right) valleys. Middle panel: linecuts of the potential along $\\phi=0$ and $\\phi=\\pi$ as indicated with white dotted lines on the image of the potential.     \n\\end{spacing}\n\n\\clearpage\n\n\\begin{figure}[htp]\n    \\centering\n    \\includegraphics[width=17cm]{figure2_v5.pdf}\n\\end{figure}\n\\begin{spacing}{1.0}\n\\noindent {\\bf Fig.~2.} \\textbf{Circuit QED with the $\\mathbf{0-\\pi}$ qubit.} (\\textbf{A}) Schematic of the capacitive coupling scheme between the qubit and the transmission-line resonator. The capacitances between the four nodes of the circuit and the resonator determine the effective coupling capacitances for the modes. (\\textbf{B}) False-color optical image of the ${0-\\pi}$ device with colors referring to the four nodes of the circuit. GND: ground plane of the resonator; $\\mathrm{V_0}$: centerpin of the resonator; JJ: Josephson junction; JJA: Josephson junction array. (\\textbf{C}) The spectroscopic response of the harmonic $\\zeta$ mode. Solid line shows the fit\\cite{masluk} with quality factors of $Q_\\mathrm{ext}^\\zeta=41,600$ and $Q_\\mathrm{int}^\\zeta=42,500$. (\\textbf{D} to \\textbf{F}) Transmission and spectroscopy measurements (background subtracted) of the ${0-\\pi}$ qubit as a function of external magnetic field (\\textbf{D}) at $n_g^\\theta=0.0\/0.5$ and (\\textbf{E}) at $n_g^\\theta=0.25$, and as a function of offset-charge bias (\\textbf{F}) at $\\Phi_\\mathrm{ext}=0$. The transmission measurements around 7.3 GHz (yellow-pink) show negligible dependence of the cavity resonance on external parameters. The spectroscopic data (green-blue) demonstrate the energy level structure of the ${0-\\pi}$ qubit, which is in excellent agreement with a coupled resonator-qubit theoretical fit (dashed lines). The result of the fit is plotted over only the positive side of the data for clarity. The low-energy fluxon transitions are not visible in the spectroscopy data due to the small dipole elements.\n\\end{spacing}\n\n\\clearpage\n\n\\begin{figure}[htp]\n    \\centering\n    \\includegraphics[width=17cm]{figure3_v5.pdf}\n\\end{figure}\n\\begin{spacing}{1.0}\n\\noindent {\\bf Fig.~3.} \\textbf{Symmetry-induced offset-charge insensitivity and drive-assisted Aharonov-Casher effect.} (\\textbf{A}) The extended potential of the ${0-\\pi}$ circuit with red dashed lines indicating the border of the phase unit cells. (\\textbf{B} and \\textbf{C}) The eigenstates are Bloch-waves $\\Psi(\\theta,\\phi)$, which are shown for the $|\\pi_{d\\theta}^\\pm\\rangle$ pair at $n_g^\\theta=0.5$. (\\textbf{E} and \\textbf{F}) The antisymmetric Wannier wavefunction $\\Phi^-(\\theta,\\phi)$ is significantly more localized than the symmetric state $\\Phi^+(\\theta,\\phi)$, leading to a decreased hopping rate between adjacent unit cells: $t^-\\ll t^+$. (\\textbf{D}) Spectroscopic measurement of the charge dispersion of the $|\\pi_{d\\theta}^\\pm\\rangle$ pair (dashed lines show the same theoretical fit as in Fig.~2D to F). The antisymmetric state has suppressed charge sensitivity compared to the symmetric state in agreement with the different hopping rates. (\\textbf{G}) Fluxon transition in the extended picture showing that a state located in the 0 valley can be excited to the valleys at $\\pm\\pi$. (\\textbf{H}) Wannier function of the initial state $|0_{p\\theta}\\rangle$ located in the 0 valley and the final state $|\\pi_{p\\theta}^-\\rangle$ located in the $\\pi$ valley or in the $-\\pi$ valley. The state in the $-\\pi$ valley has a non-zero geometric phase due to the quasiperiodic boundary conditions \\cite{supplement}. (\\textbf{I}) Autler-Townes spectroscopy between $|0_{s}\\rangle$, $|0_{p\\theta}\\rangle$ and $|\\pi_{p\\theta}^-\\rangle$ at $n_g=0.1$ [dashed lines show the fit based on Rabi splitting of levels \\cite{supplement}]. (\\textbf{J}) The extracted Rabi splitting as a function of charge bias and the theoretically expected coupling rate $g\/2\\pi$ between the levels, which demonstrates the interference pattern with $|\\cos{\\pi n_g^\\theta}|$ dependence. Error bars are estimates based on the linewidth of the transitions. \n\n\\end{spacing}\n\n\\clearpage\n\n\\begin{figure}[htp]\n    \\centering\n    \\includegraphics[width=17cm]{figure4_v5.pdf}\n\\end{figure}\n\\begin{spacing}{1.0}\n\\noindent {\\bf Fig.~4.} \\textbf{Mapping and coherent control of protected quantum states.} (\\textbf{A}) Autler-Townes spectroscopy (background subtracted) between the ground states of the $0-\\pi$ qubit using the ancillary $|\\pi_{d\\theta}^-\\rangle$ level. Inset shows the continuous drive scheme. (\\textbf{B})  Raman spectroscopy at the detuning of $\\Delta\/2\\pi=$ -30 MHz from the ancillary level [red dashed line in (A)] as a function of external magnetic field, which demonstrates a magnetic sweet spot for the disjoint ground states. (\\textbf{C} and \\textbf{D}) Coherent Rabi oscillations between the protected ground states $|0_s\\rangle$ and $|\\pi_s^{+}\\rangle$ obtained by using two, overlapping Gaussian pulses with width of 4$\\sigma$. (\\textbf{C}) The measured homodyne voltage $V_H$ as a function of drive amplitudes at a fixed detuning ($\\Delta\/2\\pi=$ -3 MHz, $\\sigma=1$ $\\mu$s), and (\\textbf{D}) $V_H$ as a function of detuning at equal drive amplitudes ($\\Omega_1=\\Omega_2$, $\\sigma=0.8$ $\\mu$s). The maximum population transfer occurs when the two drive amplitudes are equal. The dashed lines in (D) show a fit according to the effective Rabi rate of the Raman pulses. (\\textbf{E} to \\textbf{G}) Relaxation, Ramsey and spin-echo measurements of the protected $|\\pi_s^{+}\\rangle$ state, with insets showing the pulse scheme ($\\Delta\/2\\pi=$ -4 MHz  and $\\sigma=$ 200 ns). All data were taken at $n_g^\\theta=0.25$ charge bias point.\n\n\\end{spacing}\n\\end{document}\n\n\n\\section{Materials and Methods}\n\\subsection{Sample fabrication}\n\nThe device was fabricated on a 530 $\\mu$m thick, polished c-plane sapphire substrate, on which 200 nm thick niobium was sputtered using an AJA superconducting deposition system. We used optical lithography to define the resonators and shunting capacitances. AZ1505 positive photoresist was spun on the chip, baked at 95$^{\\circ}$C for 1 min and patterned using the 2 mm write-head of a Heidelberg DWL66+ tool. After developing the chip in AZ300MIF for 1 min and rinsed in running DI water for $\\sim$1 min, the sample was dry-etched in PlasmaTherm APEX SLR using the mixture of CHF$_3$, O$_2$, SF$_6$, Ar gases (with 40:1:15:10 ratios). The photoresist was stripped by Microposit Remover 1165 and solvent-cleaned by toluene, acetone, methanol, isopropanol involving sonication and a nitrogen blow-dry. For electron-beam lithography, we span MMA\/PMMA bilayer on the chip (baked for 2 + 30 min at 175$^{\\circ}$C), evaporated 40 nm thick anticharging aluminum layer, and diced the sample into single chips. We exposed the Josephson junctions in a 125 keV Elionix e-beam system (at beam current of 1 nA and aperture of 60 $\\mu$m). The anticharging layer was removed by soaking the chip in MF319 for 3 min and the e-beam resists were developed in the 1:3 mixture of methyl isobutyl ketone (MIBK) to isopropanol for 50 sec and pure isopropanol for 10 sec. The Josephson junctions were double-angle-evaporated in a Plassys e-beam-evaporator system with base pressure less than 10$^{-7}$ mbar. Before the evaporation, an \\textit{in-situ} argon ion beam etch was used to clean the surface of the sample. We evaporated 20 nm + 50 nm thick Al layers at a rate of 0.4 nm\/s and oxidized the first layer for 10 min at 200 mbar in a 15\\% oxygen-in-argon environment to realize the tunnel junction. The Al layer was lift-off in PG Remover at $\\sim$70$^{\\circ}$C and cleaned with isopropanol. \n\nThe device was placed in a copper PCB and wirebonded (\\cref{fig:device_photo}). An off-chip copper coil was attached to the PCB.  The sample holder had an aluminum shield (covered with Eccosorb CR-124 and wrapped with thin Mylar layers) and an outer mu-metal shield. The sample holder was attached to the mixing chamber plate of a dilution refrigerator with base temperature of 10 mK (\\cref{fig:fridge_wiring}).\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=17cm]{figs_device_v5.pdf}\n    \\caption{\\label{fig:device_photo}(\\textbf{A}) Optical image of a wire-bonded $0-\\pi$ device mounted into the sample holder. (\\textbf{B}) Image of the 7 mm x 7 mm chip showing the resonator with its coupling capacitors and the qubit. (\\textbf{C}) Enlarged image of the middle region of the $0-\\pi$ device, which displays the pairs of superinductors and Josephson junctions.}\n\\end{figure}\n\n\n\\subsection{Finite-element simulation of the capacitances}\n\nAs mentioned in the main text, realizing the proper capacitance values in the $0-\\pi$ circuit is a key requirement to achieve the protected regime. The large shunting capacitance in the circuit is denoted by $C$, while the cross capacitance between the nodes enclosing the superinductances (Josephson junctions) is $C_L^x$ ($C_J^x$). In our design, all four nodes are coupled to both the centerpin ($C_r^i$) and the ground plane ($C_0^i$) of the resonator (\\cref{fig:capacitance_network}). We used ANSYS Maxwell electromagnetic field simulation software to determine the capacitance values in the circuit, which are summarized in \\cref{tab:capacitance_parameters}. These parameters (with the assumptions of dielectric constant $\\epsilon_r=10.7$ for sapphire, $C_J$ = 2 fF and $E_L = $ 0.38 GHz) results in energy scales of $E_C^\\theta\/h=$ 88 MHz, $E_C^\\phi\/h=$ 1.02 GHz and $\\omega_\\zeta\/2\\pi=$ 742 MHz, which are in excellent agreement with our experimental findings.\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{ccccccccccc}\n    $C$ & $C_L^x$ & $C_J^x$ & $C_r^1$ & $C_r^2$ & $C_r^3$ & $C_r^4$ & \n    $C_0^1$ & $C_0^2$ & $C_0^3$ & $C_0^4$\\\\\\hline\\hline\n    100.5 & 0.7 & 1.0 & 9.1 & 0.3 & 3.8 & 0.3 & 8.2 & 7.9 & 6.2 & 11.6\n    \\end{tabular}\n    \\caption{\\label{tab:capacitance_parameters} Finite-element simulation of the device capacitances.  All values are given in fF units.}\n\\end{table}\n\n\\clearpage\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=17cm]{figs_fridge_v4.pdf}\n    \\caption{\\label{fig:fridge_wiring} Wiring diagram of the cryogenic- and room-temperature measurement setup.}\n\\end{figure}\n\n\\clearpage\n\n\\subsection{Spectrum fit}\n\\label{s:Spectrum Fit}\nHere we describe the multivariate fit to the experimental data based on a detailed theoretical model for the $0-\\pi$ device. We consider the circuit scheme of \\cref{fig:capacitance_network}\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=1.5]{device_capacitances.pdf}\n    \\caption{\\label{fig:capacitance_network} Full capacitance network of the $0-\\pi$ device. Red (blue) colors indicate coupling to the centerpin (ground plane) of the resonator.}\n\\end{figure}, where we have introduced additional gate ($C_r^i$) and ground ($C_0^i$) capacitances for nodes $i\\in[1,4]$. In the flux node basis $\\{\\Phi_i\\}$, the circuit Lagrangian takes the form \n\\begin{equation}\n    \\mathcal{L}_{\\Phi} = \\mathbf{\\dot{\\Phi}}^T\\cdot\\frac{\\mathbf{C}_{\\Phi}}{2}\\cdot\\mathbf{\\dot{\\Phi}} - \\mathbf{\\dot{\\Phi}}^T\\cdot\\mathbf{C}_{r}\\cdot\\mathbf{V}_{\\Phi} - U(\\mathbf{\\Phi},\\Phi_{\\mathrm{ext}}),\n    \\label{eq:circuit Lagrangian node basis}\n\\end{equation}\nwhere $\\mathbf{\\Phi}=(\\Phi_1,\\dots,\\Phi_4)^T$, $\\mathbf{C}_{\\Phi}$ is the capacitance matrix of the circuit (including gate and ground capacitances), $\\mathbf{V}_{\\Phi}= V_r (1,1,1,1)^T$ is a voltage-drive vector defined in terms of the resonator voltage, $V_r$, $\\mathbf{C}_{r}=\\mathrm{diag}(C_r^1,\\dots,C_r^4)$ is the gate capacitance matrix, and $U(\\mathbf{\\Phi},\\Phi_{\\mathrm{ext}})$ is the potential energy corresponding to the Josephson junctions and inductances of the circuit. More precisely, the circuit capacitance matrix is given by\n\\begin{equation}\n    \\mathbf{C}_{\\Phi}=\n    \\begin{pmatrix}\n        C_1 & -C_J & -C & 0 \\\\\n        -C_J & C_2 & 0 & -C \\\\\n        -C & 0 & C_3 & -C_J \\\\\n        0 & -C & -C_J & C_4 \\\\\n    \\end{pmatrix},\n    \\label{eq:full capacitance matrix}\n\\end{equation}\nwhere $C_i = C_J + C + C_{r}^i + C_{0}^i$ for $i\\in[1,4]$. We now move to the $0-\\pi$ mode basis defined by $\\mathbf{\\Theta}=(\\phi,\\theta,\\zeta,\\Sigma)^T$, by the rotation $\\mathbf{\\Theta} = \\mathbf{R}\\cdot\\mathbf{\\Phi}$,  where \n\\begin{equation}\n    \\mathbf{R}=\\frac{1}{2}\n    \\begin{pmatrix}\n        -1 & 1 & -1 & 1 \\\\\n        -1 & 1 & 1 & -1 \\\\\n        1 & 1 & -1 & -1 \\\\\n        1 & 1 & 1 & 1\\textbf{} \\\\\n    \\end{pmatrix}.\n    \\label{eq:coordinate transformation}\n\\end{equation}\nUnder such a transformation, \\cref{eq:circuit Lagrangian node basis} becomes\n\\begin{equation}\n    \\mathcal{L}_{\\Theta} = \\mathbf{\\dot{\\Theta}}^T\\cdot\\frac{\\mathbf{C}_{\\Theta}}{2}\\cdot\\mathbf{\\dot{\\Theta}} - \\mathbf{\\dot{\\Theta}}^T\\cdot\\mathbf{\\tilde{C}}_{r}\\cdot\\mathbf{V}_{\\Theta}- U(\\mathbf{\\Theta},\\Phi_{\\mathrm{ext}}),\n    \\label{eq:circuit Lagrangian mode basis}\n\\end{equation}\nwhere $\\mathbf{C}_{\\Theta} = (\\mathbf{R}^{-1})^T\\cdot\\mathbf{C}_{\\Phi}\\cdot\\mathbf{R}^{-1}$ and $\\mathbf{\\tilde{C}}_{r} = (\\mathbf{R}^{-1})^T\\cdot\\mathbf{{C}}_{r}\\cdot\\mathbf{R}^{-1}$ are the transformed capacitance matrices, and $\\mathbf{V}_{\\Theta} = \\mathbf{R}\\cdot\\mathbf{V}_{\\Phi}$ is the voltage-drive vector expressed in the $0-\\pi$ mode basis. By performing a Legendre transformation, we arrive at the circuit Hamiltonian\n\\begin{equation}\n    H = (\\mathbf{q}_{\\Theta} + \\mathbf{\\tilde{C}}_{r}\\cdot\\mathbf{V}_{\\Theta})^T\\cdot\\frac{\\mathbf{C}_{\\Theta}^{-1}}{2}\\cdot(\\mathbf{q}_{\\Theta} + \\mathbf{\\tilde{C}}_{r}\\cdot\\mathbf{V}_{\\Theta}) + U(\\mathbf{\\Theta},\\Phi_{\\mathrm{ext}}),\n    \\label{eq:circuit Hamiltonian mode basis}\n\\end{equation}\nwhere $\\mathbf{q}_{\\Theta} = {\\partial{\\mathcal{L}_{\\Theta}}}\/{{\\partial \\mathbf{\\dot{\\Theta}}}}$ is the conjugate charge vector operator. Note that \\cref{eq:circuit Hamiltonian mode basis} can be split as\n\\begin{equation}\n    H = H_{0-\\pi} + H_{\\mathrm{drive}},\n    \\label{eq:circuit Hamiltonian mode basis v2}\n\\end{equation}\nwhere \n\\begin{equation}\n    H_{0-\\pi}= \\mathbf{q}_{\\Theta}^T\\cdot\\frac{\\mathbf{C}_{\\Theta}^{-1}}{2}\\cdot\\mathbf{q}_{\\Theta} + U(\\mathbf{\\Theta},\\Phi_{\\mathrm{ext}}),\n    \\label{eq:circuit Hamiltonian mode basis ZP only}\n\\end{equation}\nis the undriven $0-\\pi$ qubit Hamiltonian and \n\\begin{equation}\n    H_{\\mathrm{drive}} = \\mathbf{q}_{\\Theta}^T\\cdot(\\mathbf{C}_{\\Theta}^{-1}\\cdot\\mathbf{\\tilde{C}}_{r})\\cdot\\mathbf{V}_{\\Theta},\n    \\label{eq:circuit Hamiltonian mode basis drive only}\n\\end{equation}\nis the drive term. \n\nWhile all circuit details are taken into account in \\cref{eq:circuit Hamiltonian mode basis v2}, the spectrum fit that is presented in the main text aims to provide the simplest possible accurate description of the device Hamiltonian. Thus, in order to simplify our treatment, we implement a few approximations. In particular, we omit any coupling to the $\\zeta$ and $\\Sigma$ modes, neglecting a potential capacitive interaction between these and the qubit modes and reducing the qubit Hamiltonian to \n\\begin{equation}\n    H_{0-\\pi} \\simeq 4E_{C}^{\\phi}n_{\\phi}^2 + 4E_{C}^{\\theta}(n_{\\theta}-n_g)^2 + \\hbar g_{\\phi\\theta} n_{\\phi}n_{\\theta} + U(\\mathbf{\\Theta},\\Phi_{\\mathrm{ext}}).\n    \\label{eq:circuit Hamiltonian mode basis ZP only reduced}\n\\end{equation}\nHere, $E_{C}^{\\phi}=e^2\/2C_{\\phi}$ and $E_{C}^{\\theta}=e^2\/2C_{\\theta}$ are the charging energies of the $\\phi$ and $\\theta$ modes and $\\hbar g_{\\phi\\theta}$ is the strength of a capacitive interaction between these modes due to the asymmetry of the circuit capacitance matrix. Accordingly, we also approximate \\cref{eq:circuit Hamiltonian mode basis drive only} by\n\\begin{equation}\n    H_{\\mathrm{drive}} \\simeq (\\beta_{\\phi}n_{\\phi} + \\beta_{\\theta}n_{\\theta})\\times2eV_{r},\n    \\label{eq:circuit Hamiltonian mode basis drive only approx}\n\\end{equation}\nwhere $\\beta_{\\phi}$ and $\\beta_{\\theta}$ are capacitive coupling ratios for the $\\phi$ and $\\theta$ modes. We moreover set $g_{\\phi\\theta}\\to 0$ in \\cref{eq:circuit Hamiltonian mode basis drive only approx}, eliminating one fit parameter. We observed, however, that deviations from $g_{\\phi\\theta}\\simeq 0$ within bounds given by finite-element estimations of the coupling capacitance do not significantly modify the quality of the fit. \n\nFor the multivariate fit, we treat all energy and coupling variables as fit parameters, including $E_{C_{\\phi}}$, $E_{C_{\\theta}}$, $\\beta_{\\phi}$, $\\beta_{\\theta}$ and those in the potential energy\n\\begin{equation}\n    U(\\mathbf{\\Theta},\\Phi_{\\mathrm{ext}}) = -2E_J\\cos\\theta\\cos(\\phi-{\\pi\\Phi_{\\mathrm{ext}}}\/{\\Phi_0}) + E_L\\phi^2 + E_J dE_J \\sin\\theta\\sin(\\phi-{\\pi\\Phi_{\\mathrm{ext}}}\/{\\Phi_0}),\n    \\label{eq:potential energy}\n\\end{equation}\ndefined in terms of the junction energy $E_J$, the superinductance energy $E_L$ and the relative junction-energy asymmetry $dE_J$. The fit also incorporates the resonator mode with nominal impedance $Z_r=50\\,\\Omega$ and frequency $f_r\\simeq 7.35\\,\\mathrm{GHz}$ parameters, for which the voltage operator reads\n\\begin{equation}\n    V_r= V_{\\mathrm{rms}} (a + a^{\\dagger}),\n    \\label{eq:Vrms}\n\\end{equation}\nwhere $V_{\\mathrm{rms}} = \\sqrt{2hf_r^2Z_r}$ for a $\\lambda\/2$ resonator, and $a$ and $a^{\\dagger}$ are the respective harmonic-oscillator ladder operators. The fit takes into account two sets of data corresponding to a sweep of the magnetic flux for the offset charges $n_g=0.0$ and $n_g=0.25$. A single error metric measures the distance between the result of the exact diagonalization of the qubit-resonator Hamiltonian and both data sets. The result of the fit is shown in \\cref{fig:spectrum_details}A and B (in addition to the figures in the main manuscript)\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=17cm]{figs_transition_data_v4.pdf}\n    \\caption{\\label{fig:spectrum_details} Spectrum fit of the $0-\\pi$ device. (\\textbf{A}) and (\\textbf{B}) correspond to the cases of $n_g=0$ and $n_g=0.25$, respectively. The experimental data used for the fit are displayed by colored circles while the theory results are given by the black dashed lines. Note that some of the qubit transitions are invisible in the experiment due to exponentially small matrix elements or vanishing dispersive shifts. The transition around $7.35\\,\\mathrm{GHz}$ corresponds to the readout resonator. (\\textbf{C}) Overlay of the transitions predicted by our model on the experimental data by assuming thermal population of the lowest two levels in the $\\pi$ valley, which explains the origin of the \\textit{bright} transitions in the spectrum ($n_g=0.25$). (\\textbf{D}) A set of cavity-assited sideband transtions that are also captured by the theory model ($n_g=0.25$).}\n\\end{figure}\nand the fit parameters are provided in \\cref{tab:fit_parameters}. These parameters are in excellent agreement with those expected from a finite-element simulation of the device.\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{ccccccc}\n    $E_{C}^{\\phi}\/h$ & $E_{C}^{\\theta}\/h$ & $E_J\/h$ & $E_L\/h$ & $dE_J$ & $\\beta_{\\phi}$ & $\\beta_{\\theta}$ \\\\\\hline\\hline\n    1.142 & 0.092 & 6.013 & 0.377 & 0.1 & $0.27$ & $6.6\\times 10^{-3}$\n    \\end{tabular}\n    \\caption{\\label{tab:fit_parameters} Result of the multivariate fit to the experimental data. All energy parameters are given in GHz units.}\n\\end{table}\n\nWe find an excellent agreement between the theoretical model and the experimental data, both for $n_g=0$ and $n_g=0.25$. As Fig.~2 of the main text shows, the obtained parameters also describe the transitions at $n_g=0.5$, and generally, the entire charge dependence of the levels. In  \\cref{fig:spectrum_details}C, we also show that additional features in the spectroscopy data can be explained by transitions between the thermally occupied fluxon states to higher levels.  Furthermore, the theoretical model not only captures accurately the qubit transitions, but also the cavity-assited sideband transitions. Since the latter transitions were not originally taken into account for the fit, this fact provides further confirmation of the validity of the theoretical model. \n\n\\section{Supplementary text}\n\n\\subsection{Tight-binding approximation}\n\nIn the main text, we introduced a tight-binding model to explain the charge dependence of the fluxon transitions and dipole matrix elements. Here, we provide additional information regarding this model. We present a bottom-up approach considering first the case of a charge-sensitive transmon, after which we focus on the case of the $0-\\pi$ qubit.\n\nTo make the connection between a charge-sensitive qubit and a periodic lattice, we briefly review the definitions of Bloch states and Wannier states in solid-state physics. The single-particle Hamiltonian describing electrons moving in a one-dimensional crystal with a periodic potential is\n\\begin{equation}\\label{crystal_Hamiltonian}\n    H_\\mathrm{crystal}= -\\frac{\\hbar^2}{2m}\\partial_x^2 + V(x),\n\\end{equation}\nwhere $m$ is the mass of the electron and $\\hbar$ is the reduced Planck constant. The eigenstates belonging to a single band are quasi-periodic Bloch states $\\Psi_k(x) = e^{ikx} u_{k}(x)$, where $u_{k}(x)$ is a lattice periodic function and $k$ is the crystal-momentum. \n\nWe define the Wannier function corresponding to a molecular orbital at a lattice site $x_0$ by \n\\begin{equation}\n    \\Phi(x-x_0) = \\frac{1}{\\sqrt{N}}\\sum_{k}e^{-ikx_0}\\Psi_{k}(x), \n\\end{equation}\nwhere $N$ is the number of sites in the lattice. An advantage of using Wannier functions is that they provide a natural choice for localized, orthonormal atomic states. The Wannier functions are non-unique due to the unconstrained phase degree of freedom of the Bloch electrons. However, there exists only one maximally localized Wannier wavefunction that is real, exponentially localized and symmetric or antisymmetric (\\textit{48}).\n\n\nWe can express the Bloch states as a function of the Wannier states corresponding to different lattice sites by an inverse Fourier transform\n\\begin{equation}\n  \\Psi_k(x) = \\frac{1}{\\sqrt{N}}\\sum_{x_0} e^{ikx_0}\\Phi(x-x_0).  \n\\end{equation}\n\nThe energy of the eigenstates as a function of momentum can be approximated by the well-known tight-binding dispersion relation of $\\epsilon_k = \\epsilon_0 + 2t\\cos{ka}$, where the energy scales are related to the localized Wannier functions as $\\epsilon_0 = \\int{dx}\\Phi^*(x)H\\Phi(x)$ and $t = \\int{dx}\\Phi^*(x-a)H\\Phi(x)$.\n\n\\paragraph{Charge-sensitive transmon}\n\nWe now consider the case of the offset-charge sensitive transmon qubit\n\\begin{equation}\n    H= 4E_C(i\\partial_\\theta-n_g)^2-E_J\\cos\\theta,\n\\end{equation}\nwhere $E_C$ is the charging energy and $E_J$ is the Josephson energy. The $p$th eigenstate of this Hamiltonian for a given $n_g$ obeys \n\\begin{equation}\nHu_{n_g}^p(\\theta)=\\epsilon_{n_g}^p u_{n_g}^p(\\theta),\n\\end{equation}\nwhere $u_{n_g}^p(\\theta)$ is a $2\\pi$-periodic function of the superconducting phase across the Josephson junction, i.e.,\n\\begin{equation}\n    u^p_{n_g}(\\theta) = u^p_{n_g}(\\theta + 2\\pi).\n\\end{equation}\nNext, we perform a gauge transformation defined by the unitary  $U=e^{in_g\\theta}$, in order to eliminate the offset-charge dependence of the transmon Hamiltonian which becomes\n\\begin{equation}\\label{transmon_Hamiltonian_periodic}\n    \\bar{H}= -4E_C\\partial_\\theta^2-E_J\\cos\\theta,\n\\end{equation}\nand thus\n\\begin{equation}\\label{transmon_wavefunction_quasiperiodic}\n    \\Psi^p_{n_g}(\\theta) = e^{in_g\\theta} u^p_{n_g}(\\theta).\n\\end{equation}\nWe note that, in this gauge, the transmon Hamiltonian [\\cref{transmon_Hamiltonian_periodic}] is identical to the Hamiltonian of a one-dimensional crystal [\\cref{crystal_Hamiltonian}]. Therefore, the eigenstates of \\cref{transmon_Hamiltonian_periodic} are also quasi-periodic Bloch waves.\n\nIn analogy to the solid state case, we introduce the Wannier functions for the transmon qubit by Fourier transforming the Bloch states in $n_g$\n\\begin{equation}\n    \\Phi_p(\\theta-2\\pi l) = \\frac{1}{\\sqrt{N}}\\sum_{n_g}e^{-i2\\pi l\\cdot n_g}\\Psi^p_{n_g}(\\theta), \n\\end{equation}\nwhere $l$ is an integer that corresponds to the number of the unit cell where the Wannier function is localized. The inverse Fourier transform thus reads\n\\begin{equation}\n    \\Psi_{n_g}^p(\\theta) = \\frac{1}{\\sqrt{N}}\\sum_{\\theta_0}e^{i2\\pi l\\cdot n_g}\\Phi_p(\\theta-2\\pi l). \n\\end{equation}\n\nThe $n_g$ dispersion of the low-lying transmon energy levels is determined by the hopping matrix element: $\\epsilon_{n_g}^p = \\epsilon_0^p + 2t^p\\cos{2\\pi n_g}$, where $\\epsilon_0^p = \\int{d\\theta}\\Phi^{p*}(\\theta)H\\Phi^p(\\theta)$ and $t^p = \\int{d\\theta}\\Phi^{p*}(\\theta-2\\pi)H\\Phi^p(\\theta)$.\n\nNext, we consider the charge matrix element $n_{pq} = \\langle u^p | i\\partial_\\theta | u^q \\rangle$ in the tight-binding approximation. Since the states are assumed to be localized, we only consider the\ncontribution of two states which are in the same well or nearest neighbours, thus\n\\begin{align}\n    \\langle u^p | i\\partial_\\theta | u^q \\rangle =\\, & i\\int d\\theta \\Phi_p^\\ast(\\theta) \\partial_\\theta \\big(\\Phi_q(\\theta) + \\Phi_q(\\theta+2\\pi)e^{-i 2\\pi n_\\text{g}} + \\Phi_q(\\theta-2\\pi)e^{i 2\\pi n_\\text{g}}) \\big)\\notag \\\\\n    & + n_\\text{g}\\int d\\theta \\Phi_p^\\ast(\\theta)\\big(\\Phi_q(\\theta) + \\Phi_q(\\theta+2\\pi)e^{-i 2\\pi n_\\text{g}} + \\Phi_q(\\theta-2\\pi)e^{i 2\\pi n_\\text{g}} \\big),\n\\end{align}\nand for further reference, we define the following variables\n\\begin{gather*}\n    \\eta_0^\\text{C} = \\int d\\theta \\Phi_p^\\ast(\\theta) \\Phi_q(\\theta), \\qquad \\eta_0^\\text{L} = \\int d\\theta \\Phi_p^\\ast(\\theta) \\Phi_q(\\theta+2\\pi), \\\\ \\eta_0^\\text{R} = \\int d\\theta \\Phi_p^\\ast(\\theta) \\Phi_q(\\theta-2\\pi), \\qquad\n    \\eta_1^\\text{C} = i\\int d\\theta \\Phi_p^\\ast(\\theta)\\partial_\\theta \\Phi_q(\\theta), \\\\\n    \\eta_1^\\text{L} = i\\int d\\theta \\Phi_p^\\ast(\\theta)\\partial_\\theta \\Phi_q(\\theta+2\\pi), \\qquad \\eta_1^\\text{R} = i\\int d\\theta \\Phi_p^\\ast(\\theta)\\partial_\\theta \\Phi_q(\\theta-2\\pi).\n\\end{gather*}\n\nThe matrix element can be written as \n\\begin{equation}\n    \\langle u^p | i\\partial_\\theta | u^q \\rangle =\\, (\\eta_1^\\text{C} +\\eta_1^\\text{L}e^{-i 2\\pi n_\\text{g}} + \\eta_1^\\text{R}e^{i 2\\pi n_\\text{g}} ) + n_\\text{g}(\\eta_0^\\text{C} +\\eta_0^\\text{L}e^{-i 2\\pi n_\\text{g}} + \\eta_0^\\text{R}e^{i 2\\pi n_\\text{g}}).\n\\end{equation}\nWe now consider the case where $\\Phi_p$ and $\\Phi_q$ have the same parity. It follows then, that $\\eta_0^\\text{C}=\\eta_1^\\text{C}=0$, $\\eta_1^\\text{L}=-\\eta_1^\\text{R}$, and $\\eta_0^\\text{L}=\\eta_0^\\text{R}$. The matrix element thus simplifies to \n\\begin{equation}\n    |\\langle u^p | i\\partial_\\theta | u^q \\rangle| = |-2 i \\eta_1^\\text{L}\\sin{2\\pi n_\\text{g}} + 2n_\\text{g}\\eta_0^\\text{L}\\cos{2\\pi n_\\text{g}}|.\n\\end{equation}\nSince the states are all localized in the corresponding wells, the part that contributes to the integral is the tail of the wavefunction. Assuming the tail is of Gaussian type $\\exp(-\\theta^2)$, we have $|\\eta_1^\\text{L}|\\gg|\\eta_0^\\text{L}|$, and the matrix element is further simplified \n\\begin{equation}\n    |\\langle u^p | i\\partial_\\theta | u^q \\rangle| = |2\\eta_1^\\text{L}\\sin{2\\pi n_\\text{g}}|.\n\\end{equation}\n\nThe case in which $\\Phi_p$ and $\\Phi_q$ have opposite parities leads to $\\eta_0^\\text{C}=0$, $\\eta_1^\\text{L}=\\eta_1^\\text{R}$, and $\\eta_0^\\text{L}=-\\eta_0^\\text{R}$. The matrix element is then\n\\begin{equation}\n    |\\langle u^p | i\\partial_\\theta | u^q \\rangle| = |\\eta_1^\\text{C} + 2\\eta_1^\\text{L}\\cos{2\\pi n_\\text{g}} - 2 i n_\\text{g}\\eta_0^\\text{L}\\sin{2\\pi n_\\text{g}}|.\n\\end{equation}\nTaking into account $|\\eta_1^\\text{C}|\\gg|\\eta_1^\\text{L}|\\gg|\\eta_0^\\text{L}|$, we have \n\\begin{equation}\n    |\\langle u^p | i\\partial_\\theta | u^q \\rangle| = |\\eta_1^\\text{C} + 2\\eta_1^\\text{L}\\cos{2\\pi n_\\text{g}}|.\n\\end{equation}\n\n\\paragraph{Charge-sensitivity in the $0-\\pi$ qubit}\n\nThe charge sensitivity in the $0-\\pi$ qubit enters through the $\\theta$ mode as\n\\begin{align}\n    H_{0-\\pi}=4E_C^\\theta(n_\\theta - n_g^\\theta)^2 +4E_C^\\phi n_\\phi^2 + V(\\theta,\\phi).\n\\end{align}\nThe $u_p(\\theta,\\phi)$ eigenstates of the $0-\\pi$ Hamiltonian are $2\\pi$-periodic in $\\theta$ as $u_p(\\theta,\\phi)=u_p(\\theta+2\\pi,\\phi)$. We define the Bloch states again as $\\Psi^p_{n_g}(\\theta,\\phi) = e^{in_g\\theta} u_p(\\theta,\\phi)$ and the Wannier states as $\\Phi_p(\\theta-2\\pi l,\\phi) = \\frac{1}{\\sqrt{N}}\\sum_{n_g}e^{-i2\\pi l\\cdot n_g}\\Psi^p_{n_g}(\\theta,\\phi)$, where $l$ is an integer.\n\nThe low-energy levels are either localized in the 0 or in the $\\pi$-valley, which we here explicitly note by a superscript: $u_p^0(\\theta,\\phi)$ and $u_p^\\pi(\\theta,\\phi)$. We also distinguish between Wannier wavefunctions localized in the two different valleys by introducing \n\\begin{align}\n    \\Phi_p^0(\\theta,\\phi)= \\Phi_p(\\theta,\\phi), \\\\\n    \\Phi_p^\\pi(\\theta,\\phi)= \\Phi_p(\\theta+\\pi,\\phi),\n\\end{align}\nwhich are centered around the center of the 0 and $\\pi$ valley, respectively.\n\nNext we consider the charge matrix elements $\\langle u_p^0 | i\\partial_\\theta | u_q^\\pi \\rangle$ and $\\langle u_p^0 | i\\partial_\\phi | u_q^\\pi \\rangle$. Since the states are assumed to be localized either in the $\\theta=0$ or $\\theta=\\pi$ well, we only consider the contribution of two states being the nearest neighbours, i.e. for $l$ component of $u_p^0$, we only keep $l$, $l+1$ components of $u_q^\\pi$. The matrix elements are then\n\\begin{align}\n    \\langle u_p^0 | i\\partial_\\theta | u_q^\\pi \\rangle =\\, & i\\int d\\theta d\\phi\\, \\Phi_p^0(\\theta,\\phi) \\partial_\\theta \\big(\\Phi_q^\\pi(\\theta-\\pi,\\phi) + \\Phi_q^\\pi(\\theta+\\pi,\\phi)e^{-i 2\\pi n_\\text{g}}  \\big) \\notag \\\\ \n    & + n_\\text{g}\\int d\\theta d\\phi \\, \\Phi_p^0(\\theta,\\phi)\\big(\\Phi_q^\\pi(\\theta-\\pi,\\phi) + \\Phi_q^\\pi(\\theta+\\pi,\\phi)e^{-i 2\\pi n_\\text{g}}  \\big), \\\\\n    \\langle u_p^0 | i\\partial_\\phi | u_q^\\pi \\rangle =\\, & i\\int d\\theta d\\phi\\, \\Phi_p^0(\\theta,\\phi) \\partial_\\phi \\big(\\Phi_q^\\pi(\\theta-\\pi,\\phi) + \\Phi_q^\\pi(\\theta+\\pi,\\phi)e^{-i 2\\pi n_\\text{g}}  \\big).\n\\end{align}\nFor simplicity, we define the following variables\n\\begin{align*}\n    \\eta_0^\\text{L} &= \\int d\\theta d\\phi \\Phi_p^0(\\theta,\\phi) \\Phi_q^\\pi(\\theta+\\pi,\\phi), \\qquad \\eta_0^\\text{R} = \\int d\\theta d\\phi  \\Phi_p^0(\\theta,\\phi) \\Phi_q^\\pi(\\theta-\\pi,\\phi),\\\\\n    \\eta_1^\\text{L} &= i\\int d\\theta d\\phi \\Phi_p^0(\\theta,\\phi)\\partial_\\theta \\Phi_q^\\pi(\\theta+\\pi,\\phi), \\qquad \\eta_1^\\text{R} = i\\int d\\theta d\\phi \\Phi_p^0(\\theta,\\phi)\\partial_\\theta \\Phi_q^\\pi(\\theta-\\pi,\\phi), \\\\\n    \\eta^\\text{L} &= i\\int d\\theta d\\phi \\Phi_p^0(\\theta,\\phi)\\partial_\\phi \\Phi_q^\\pi(\\theta+\\pi,\\phi), \\qquad \\eta^\\text{R} = i\\int d\\theta d\\phi \\Phi_p^0(\\theta,\\phi)\\partial_\\phi \\Phi_q^\\pi(\\theta-\\pi,\\phi). \n\\end{align*}\nThe matrix element thus can be written as \n\\begin{align}\n    \\langle u_p^0 | i\\partial_\\theta | u_q^\\pi \\rangle &=\\, (\\eta_1^\\text{L}e^{-i 2\\pi n_\\text{g}} +\\eta_1^\\text{R}) + n_\\text{g}(\\eta_0^\\text{L}e^{-i 2\\pi n_\\text{g}} +\\eta_0^\\text{R}), \\\\\n    \\langle u_p^0 | i\\partial_\\phi | u_q^\\pi \\rangle &=\\, \\eta^\\text{L}e^{-i 2\\pi n_\\text{g}}  +\\eta^\\text{R} .\n\\end{align}\n\nThe dipole matrix elements can be further simplified depending on the parities along the $\\theta$ and $\\phi$ directions, which eventually leads to the charge dependence of the fluxon dipole matrix elements summarized in \\cref{tab:fluxon_transitions}. The plasmon transition in the $0-\\pi$ qubit is similar to the case of the transmon qubit, and the result is summarized in the \\cref{tab:plasmon_transitions}.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{ |c|c|c|c|c| } \n \\hline\n  & \\thead{$\\Pi_i^\\theta = \\Pi_j^\\theta$, \\\\ $\\Pi_i^\\phi = \\Pi_j^\\phi $ } & \\thead{$\\Pi_i^\\theta =- \\Pi_j^\\theta$, \\\\ $\\Pi_i^\\phi = \\Pi_j^\\phi $} & \\thead{$\\Pi_i^\\theta = \\Pi_j^\\theta$,\\\\ $\\Pi_i^\\phi =- \\Pi_j^\\phi $} & \\thead{$\\Pi_i^\\theta = -\\Pi_j^\\theta$, \\\\ $\\Pi_i^\\phi = -\\Pi_j^\\phi $}  \\\\ \n  \\hline\n  $|\\langle i |i\\partial_\\theta | j \\rangle|$ & $|\\sin{\\pi n_\\text{g}}|$ & $|\\cos{\\pi n_\\text{g}}|$ & 0 & 0   \\\\ \n  \\hline\n  $|\\langle i |i\\partial_\\phi | j \\rangle|$ & 0 & 0 & $|\\cos{\\pi n_\\text{g}}|$ & $|\\sin{\\pi n_\\text{g}}|$ \\\\ \n \\hline\n\\end{tabular}\n\\caption{\\label{tab:fluxon_transitions}Matrix elements for fluxon transition.}\n\\label{table2}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{ |c|c|c|c|c| } \n \\hline\n  & \\thead{$\\Pi_i^\\theta = \\Pi_j^\\theta$, \\\\ $\\Pi_i^\\phi = \\Pi_j^\\phi $}  & \\thead{$\\Pi_i^\\theta =- \\Pi_j^\\theta$, \\\\ $\\Pi_i^\\phi = \\Pi_j^\\phi $} & \\thead{$\\Pi_i^\\theta = \\Pi_j^\\theta$, \\\\ $\\Pi_i^\\phi =- \\Pi_j^\\phi $} & \\thead{$\\Pi_i^\\theta = -\\Pi_j^\\theta$, \\\\ $\\Pi_i^\\phi = -\\Pi_j^\\phi $}  \\\\ \n  \\hline\n  $|\\langle i |i\\partial_\\theta | j \\rangle|$ & $|\\sin{2\\pi n_\\text{g}}|$ & $|1+\\epsilon\\cos{2\\pi n_\\text{g}}|$ & 0 & 0   \\\\ \n  \\hline\n  $|\\langle i |i\\partial_\\phi | j \\rangle|$ & 0 & 0 & $|1+\\epsilon\\cos{2\\pi n_\\text{g}}|$ & $|\\sin{2\\pi n_\\text{g}}|$ \\\\ \n \\hline\n\\end{tabular}\n\\caption{\\label{tab:plasmon_transitions}Matrix elements for plasmon transition.}\n\\label{table1}\n\\end{table}\n\n\\subsection{Population transfer in the two-tone Raman pulse scheme}\nWe model the Raman pulse scheme in the $0-\\pi$ qubit by truncating the energy level structure to the ground states of the valleys $|0_s\\rangle$, $|\\pi_s^+\\rangle$ and the intermediate level $|\\pi_{d\\theta}^-\\rangle$. For simplicity, we relabel these levels by $|0_s\\rangle\\to|0\\rangle$, $|\\pi_s^+\\rangle\\to|2\\rangle$ and $|\\pi_{d\\theta}^-\\rangle\\to|1\\rangle$ (see \\cref{fig:Raman_pulses}A). We first consider the unitary evolution of this $\\Lambda$-system driven by two classical fields (\\textit{44})\n\\begin{equation}\n    H\/\\hbar = \\omega_1 \\sigma_{11} + \\omega_2 \\sigma_{22} + \\left[\\Omega_\\alpha \\cos{\\left(\\omega_\\alpha t\\right)} \\sigma_{01} + \\Omega_\\beta\\cos{\\left(\\omega_\\beta t\\right)} \\sigma_{12}  + h.c.\\right],\n    \\label{eq:HLambda}\n\\end{equation}\nwhere $\\omega_0=0<\\omega_2<\\omega_1$ are the eigenfrequencies of $|0\\rangle$, $|2\\rangle$ and $|1\\rangle$, respectively, $\\omega_\\alpha$ and $\\omega_\\beta$ are the frequencies of the drive tones with amplitudes $\\Omega_\\alpha$ and $\\Omega_\\beta$, respectively, while $\\sigma_{ij} = |i \\rangle \\langle j|$ for $i,j\\in[1,2,3]$. We moreover assume that the $\\alpha$ ($\\beta$) drive addresses only the $|0\\rangle\\leftrightarrow |1\\rangle$ ($|1\\rangle\\leftrightarrow|2\\rangle$) transition.\n\nMoving to a rotating frame where the drives are equally detuned from the ancillary level $|1\\rangle$, i.e. $\\omega_\\alpha = \\omega_1 - \\Delta$, $\\omega_\\beta=\\omega_1-\\omega_2-\\Delta$, and performing the RWA approximation, \\cref{eq:HLambda} takes the time-independent form of \n\\begin{equation}\n    \\Tilde{H}\/\\hbar = \\Delta\\sigma_{11} + \\left[\\frac{1}{2}\\Omega_\\alpha\\sigma_{01} + \\frac{1}{2}\\Omega_\\beta\\sigma_{12}+h.c.\\right].\n    \\label{eq:HLambdarot}\n\\end{equation}\nDefining $\\Tilde\\Omega=\\sqrt{\\Delta^2 + \\Omega_\\alpha^2 + \\Omega_\\beta^2}$, the eigenfrequencies of \\cref{eq:HLambdarot} are given by \n\\begin{equation}\n\\begin{split}\n    \\epsilon_0 &= 0,\\\\\n    \\epsilon_\\pm &= \\frac{1}{2}\\left(\\Delta\\pm \\Tilde\\Omega\\right),\n\\end{split}\n\\end{equation}\nand correspond to the dressed states \n\\begin{equation}\n\\begin{split}\n    |\\Psi_0\\rangle &= -\\Omega_\\beta|0\\rangle + \\Omega_\\alpha|2\\rangle,\\\\\n    |\\Psi_\\pm\\rangle &=\\Omega_\\alpha|0\\rangle+(\\Delta\\pm\\Tilde\\Omega)|1\\rangle + \\Omega_\\beta|2\\rangle,\n\\end{split}\n\\end{equation}\nrespectively.\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=17cm]{figs_Raman_Rabi_model_v4.pdf}\n    \\caption{\\label{fig:Raman_pulses} (\\textbf{A}) Schematic representation of three $0-\\pi$ qubit levels coupled to two microwave drives forming a $\\Lambda$-system. (\\textbf{B} to \\textbf{D}) Level population as a function of the drive amplitudes based on the analytical results of the time-evolution of the system ($t=6.7\\ \\mu$s, $\\Delta \/ 2 \\pi$ = 20 MHz). (\\textbf{E}) Level population as a function of time based on the exact results for the effective two level system ($\\Omega_\\alpha \/ 2 \\pi=\\Omega_\\beta \/ 2 \\pi=$ 5 MHz, $\\Delta \/ 2 \\pi$ = 20 MHz). (\\textbf{F} to \\textbf{H}) Results of the numerical simulation for two Gaussian pulses with $\\sigma=1\\ \\mu$s, $\\Delta \/ 2 \\pi$ = 3 MHz, relaxation rates of $\\Gamma_{10} \/ 2 \\pi= \\Gamma_{12}\/ 2 \\pi =$ 100 kHz and dephasing rate of $\\Gamma_1^\\phi \/ 2 \\pi=$ 500 kHz.}\n\\end{figure}\n\nWe assume that the system at $t =0$ is initialized in the $|0\\rangle$ state when the drives are instantaneously turned on (square pulse). The time-evolution of the system can be obtained by using a basis transformation into the dressed basis. The system evolves to the state $|\\Psi(t)\\rangle = \\alpha(t)|0\\rangle + \\beta(t)|1\\rangle + \\gamma(t)|2\\rangle$ at time $t$, where\n\\begin{equation}\n\\begin{split}\n    \\alpha(t) &= \\frac{\\Omega_\\alpha^2}{\\Omega_\\alpha^2 + \\Omega_\\beta^2}\\times\\left[\\frac{\\Omega_\\beta^2}{\\Omega_\\alpha^2}+e^{-i\\Delta t\/2}\\left(\\cos\\frac{\\Tilde\\Omega t}{2}+i\\frac{\\Delta}{\\Tilde\\Omega}\\sin\\frac{\\Tilde\\Omega t}{2}\\right)\\right],\\\\\n    \\beta(t) &=\\frac{\\Omega_\\alpha}{\\Tilde\\Omega}\\times\\left[-ie^{-i\\Delta t\/2}\\sin\\frac{\\Tilde\\Omega t}{2} \\right],\\\\\n    \\gamma(t) &= \\frac{\\Omega_\\alpha\\Omega_\\beta}{\\Omega_\\alpha^2 + \\Omega_\\beta^2}\\times\\left[-1+e^{-i\\Delta t\/2}\\left(\\cos\\frac{\\Tilde\\Omega t}{2}+i\\frac{\\Delta}{\\Tilde\\Omega}\\sin\\frac{\\Tilde\\Omega t}{2}\\right)\\right].\n\\end{split}\n\\end{equation}\n\n\\cref{fig:Raman_pulses}E shows the level populations as a function of time for $\\Omega_1=\\Omega_2$. We observe Rabi oscillations between the two ground states $|0\\rangle$ and $|2\\rangle$ with only a negligible population in the intermediate level $|1\\rangle$. Interestingly, the Rabi oscillation features a superimposed low amplitude, high frequency modulation (\\textit{44}). We note that adiabatic elimination of the intermediate level in the vicinity of equal drives $\\Omega_1\\approx\\Omega_2$ leads to an effective two-level system (\\textit{43}) with Rabi rate of $\\Omega_R=\\Omega_1\\Omega_2\/2\\Delta$. This effective model is in good agreement with the exact analytical solution (dashed and dotted lines in \\cref{fig:Raman_pulses}E).\n\n\\cref{fig:Raman_pulses}B to D show the level population at a given time as function of the drive amplitude and detuning, similar to the pulsed measurements carried out in our experiment. The results show that maximal population transfer between $|0\\rangle$ and $|2\\rangle$ is possible when the drives are equal. \n\nAdditionally to the exact solutions, we carried out numerical simulations using the QuTiP software package (\\textit{47}) to solve the time evolution of the system involving Gaussian-shaped pulses and decay mechanisms using a Lindblad Master-equation solver. The result of the numerical simulation is in very good agreement with our experimental findings, see Fig.~4C and \\cref{fig:Raman_pulses}F to H.\n\n\\subsection{Coherence times as a function of external flux}\n\nWe mapped out the flux-dependence of the coherence times of the logical qubit states in the close vicinity of $\\Phi_\\mathrm{ext}=0$ (\\cref{fig:coherence_flux}). The data demonstrate that the Ramsey coherence times have strong dependence on the magnetic flux with a significant enhancement around the sweet spot. \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=6cm]{figs_coherence_flux_v4.pdf}\n    \\caption{\\label{fig:coherence_flux} Measured $T_{2R}$ and $T_{2E}$ values around the magnetic sweet spot. Dashed lines are a guide to the eye.}\n\\end{figure}\n\n\\subsection{Autler-Townes spectroscopy as a function of the offset charge}\n\nFor completeness, we report all measured Autler-Townes spectroscopy maps obtained at different offset-charge bias, in addition to the one presented in Fig.~3I. As discussed in the main text, we use a strong drive to dress the $|0_{p\\theta}\\rangle\\leftrightarrow|\\pi_{p\\theta}^-\\rangle$ transition. Denoting the qubit transition by $\\omega_q$, the coupling rate by $\\Omega_c$ and the coupler drive frequency by $\\omega_c$, the dispersion of the dressed states takes the form of $\\epsilon_\\pm=(\\omega_q - \\omega_c)\\pm\\sqrt{(\\omega_q - \\omega_c)^2 + \\Omega_c^2}$, which can be measured by an additional weak probe tone. \n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=17cm]{figs_AT_all_scans_v4.pdf}\n    \\caption{\\label{fig:AT_scans} Raw data of the Autler-Townes spectroscopy as a function of the offset-charge. Black and red dashed lines show the least-squares fit to the data in case of odd and even charge parity.}\n\\end{figure}\n\n\n\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nGoogle devised PageRank to help determine the importance of nodes in a directed graph representing web pages~\\cite{page1999-pagerank}. Given a random walk on a directed graph, the PageRank modification builds a new Markov chain that always has a unique stationary distribution. This new random walk models a ``random surfer'' that, with probability $\\alpha < 1$ takes a step according to the Markov chain and with probability $1-\\alpha$ randomly jumps according to a fixed distribution. If $\\mP$ is a \\emph{column stochastic} matrix that represents the random walk on the original graph, then the PageRank vector $\\vx$ is unique and solves the linear system: \n\\[ \\vx = \\alpha \\mP \\vx + (1-\\alpha) \\vv, \\]\nwhere $\\vv$ is a stochastic vector and $\\alpha$ is a probability (Section~\\ref{sec:pr} has a formal derivation). The simple Richardson iteration even converges fast for the values of $\\alpha$ that are used in practice. \n\nAlthough Google described PageRank for the web graph, the same methodology has been deployed in many applications where the importance of nodes provides insight into an underlying phenomena represented by a graph~\\cite{morrison2005-generank,freschi2007-proteinrank,Winter-2012-CancerRank,Gleich-preprint-pagerank-beyond}. We find the widespread success of the PageRank methodology intriguing and believe that there are a few important features that contributed to PageRank's success. First and second are the uniqueness and fast convergence. These properties enable reliable and efficient evaluation of the important nodes. Third, in most applications of PageRank, the input graph may contain modeling or sampling errors, and thus, PageRank's jumps are a type of regularization. This may help capture important features in the graph despite the noise.\n\nIn this paper, we begin by developing the PageRank modification to a higher-order Markov chain (Section~\\ref{sec:hopr}). These higher-order Markov chains model stochastic processes that depend on more history than just the previous state. (We review them formally in Section~\\ref{sec:homc}.) In a second-order chain, for instance, the choice of state at the next time step depends on the last two states. However, this structure corresponds to a first-order, or traditional, Markov chain on a tensor-product state-space. We show that higher-order PageRank enjoys the same uniqueness and fast convergence as in the traditional PageRank problem (Theorem~\\ref{thm:hopr}); although computing these stationary distributions is prohibitively expensive in terms of memory requirements. \n\nRecent work by~\\citet{Li-2013-tensor-markov-chain} provides an alternative: they consider a rank-1 approximation of these distributions. When we combine the PageRank modification of a higher-order Markov chain with the Li--Ng approximation, we arrive at the \\emph{multilinear PageRank problem} (Section~\\ref{sec:tensor-pr}).\nFor the specific case of an $n$-state second-order Markov chain, described by an $n \\times n \\times n$ transition probability table, the problem becomes finding the solution $\\vx$ of the polynomial system of equations: \n\\[ \\vx = \\alpha \\mR (\\vx \\kron \\vx) + (1-\\alpha) \\vv, \\]\nwhere $\\mR$ is an $n \\times n^2$ column stochastic matrix (that represents the probability table), $\\alpha$ is a probability, $\\kron$ is the Kronecker product, and $\\vv$ is a probability distribution over the $n$-states encoded as an $n$-vector.  We have written the equations in this way to emphasize the similarity to the standard PageRank equations. \n\nOne of the key contributions of our work is that the solution $\\vx$ has an interpretation as the stationary distribution of a process we describe and call the ``spacey random surfer.'' The spacey random surfer continuously forgets its own immediate history, but does remember the aggregate history and combines the current state with this aggregate history to determine the next state (Section~\\ref{sec:process}). This process provides a natural motivation for the multilinear PageRank vector in relationship to the PageRank random surfer. We build on recent advances in vertex reinforced random walks~\\cite{Pemantle-1992-vertex-reinforced,Benaim-1997-vrrw} in order to make this relationship precise.\n\n\nThere is no shortage of data analysis methods that involve tensors. These usually go by taking an $m$-way array as an order-$m$ tensor and then performing a tensor decomposition. When $m = 2$, this is often the matrix SVD and the factors obtained give the directions of maximal variation. When $m > 2$, the solution factors lose this interpretation. Understanding the resulting decompositions may be problematic without an identifiability result such as \\citet{Anandkumar-preprint-topics}. Our proposal differs in that our tensor represents a probability table for a stochastic process and, instead of a decomposition, we seek an eigenvector that has a natural interpretation as a stationary distribution.   In fact, a general non-negative tensor can be regarded as a contingency table, which can be converted into a multidimensional probability table. These tables may be regarded as the probability distribution of a higher-order Markov chain, just like how a directed graph becomes a random walk. Given the breadth of applications of tensors, our motivation was that the multilinear PageRank vector would be a unique, meaningful stationary distribution that we could compute quickly.\n\nMultilinear PageRank solutions, however, are more complicated . They are not unique for any $\\alpha < 1$ as was the case for PageRank, but only when $\\alpha < 1\/(m-1)$ where $m-1$ is the order of the Markov chain (or $m$ is the order of the underlying tensor) as shown in Theorem~\\ref{thm:tpr-unique}. We then consider five algorithms to solve the multilinear PageRank system: a fixed-point method, a shifted fixed-point method, a nonlinear inner-outer iteration, an inverse iteration, and a Newton iteration (Section~\\ref{sec:methods}). These algorithms are all fast in the unique regime. Outside that range, we used exhaustive enumeration and random sampling to build a repository of problems that do not converge with our methods. Among the challenging test cases, the inner-outer algorithm and Newton's method has the most reliable convergence properties (Section~\\ref{sec:experiments}). Our codes are available for others to use and to reproduce the figures of this manuscript: \\url{https:\/\/github.com\/dgleich\/mlpagerank}.\n\n\n\n\\section{Background} \\label{sec:background}\nThe technical background for our paper includes a brief review of Li and Ng's factorization of the stationary distribution of a higher-order PageRank Markov chain, which we discuss after introducing our notation.\n\n\\subsection{Notation}\nMatrices are bold, upper-case Roman letter, as in $\\mA$; vectors are bold, lower-case Roman letters, as in $\\vx$; and tensors are bold, underlined, upper-case Roman letters, as in $\\cmP$. We use $\\ve$ to be the vector all ones. Individual elements such as $A_{ij}$, $x_i$, or $\\ensuremath{\\cel{P}}_{ijk}$ are always written without bold-face. In some of our results, using subscripts is sub-optimal, and we will use Matlab indexing notation instead $A(i,j)$, $x(i)$, or $\\ensuremath{\\cel{P}}(i,j,k)$. An order-$m$, $n$-dimensional tensor has $m$ indices that range from $1$ to $n$. We will use $\\kron$ to denote the Kronecker product. Throughout the paper, we call a nonnegative matrix $\\mA$ column-stochastic if $\\sum_i A_{ij} = 1$. A stochastic tensor is a tensor that is nonnegative and where the sum over the first index $i$ is $1$.  We caution our readers that what we call a ``tensor'' in this article really should be called a \\textit{hypermatrix}, that is, a specific coordinate representation of a tensor. See \\citet{Lim-2014-tensors} for a discussion of the difference between a tensor and its coordinate representation.\n\n \n \n\n\nWe use $S_1, S_2, \\dots$ to denote a discrete time stochastic process on the state space $1, \\dots, n$. The probability of an event is denoted $\\Pr ( S_t = i )$ and $\\Pr( S_t = i \\mid S_{t-1} = j )$ is the conditional probability of the event. (For those experts in probability, we use this simplifying notation instead of the natural filtration given the history of the process.)\n\n\\subsection{PageRank}\n\\label{sec:pr}\nIn order to justify our forthcoming use of the term higher-order PageRank, we wish to precisely define a PageRank problem and PageRank vector. The following definition captures the discussions in \\citet{langville2006-book}.\n\\begin{definition}[PageRank] \\label{def:pr} \nLet $\\mP$ be a column stochastic matrix, let $\\alpha$ be a probability smaller than $1$, and let $\\vv$ be a stochastic vector. A PageRank vector $\\vx$ is the \\emph{unique} solution of the linear system: \n\\begin{equation} \\label{eq:pr}\n \\vx = \\alpha \\mP \\vx + (1-\\alpha) \\vv. \n\\end{equation}  We call the set $(\\alpha, \\mP, \\vv)$ a PageRank problem.\n\\end{definition}\n\nNote that the PageRank vector $\\vx$ is equivalently a Perron vector of the matrix: \n\\[ \\mM = \\alpha \\mP + (1-\\alpha) \\vv \\ve^{T} \\]\nunder the normalization that $\\vx \\ge 0$ and $\\ve^{T} \\vx = 1$.  The matrix $\\mM$ is column stochastic and encodes the behavior of the random surfer that, with probability $\\alpha$, transitions according to the Markov chain with transition matrix $\\mP$, and, with probability $(1-\\alpha)$ ``teleports'' according to the fixed distribution $\\vv$. When PageRank is used with a graph, then $\\mP$ is almost always defined as the random walk transition matrix for that graph. If a graph does not have a valid transition matrix, then there are a few adjustments available to create one~\\cite{boldi2007-traps}. \n\nWhen we solve for $\\vx$ using the power method on the Markov matrix $\\mM$ or the Richardson iteration on the linear system~\\eqref{eq:pr}, then we iterate: \n\\[ \\vx\\itn{0} = \\vv \\qquad \\vx\\itn{k+1} = \\alpha \\mP \\vx\\itn{k} + (1-\\alpha) \\vv. \\]\nThis iteration satisfies the error bound\n\\[ \\normof[1]{\\vx\\itn{k} - \\vx} \\le 2 \\alpha^k \\]\nfor any stochastic $\\vx\\itn{0}$. For values of $\\alpha$ between $0.5$ and $0.99$, which occur most often in practice, this simple iteration converges quickly.\n\n\\subsection{Higher-order Markov chains}\n\\label{sec:homc}\nWe wish to extend PageRank to higher-order Markov chains and so we briefly review their properties.\nAn $m$\\textsuperscript{th}-order Markov chain $S$ is a stochastic process that satisfies:\n\\[\n\\begin{aligned}\n\\Pr(S_t=i_1 \\mid S_{t-1}=i_2, \\dots, S_1 = i_t)\n=\\Pr(S_t = i_1 \\mid S_{t-1}=i_2, \\dots, S_{t-m} = i_{m+1}).\n\\end{aligned}\n\\]\nIn words, this means that the future state only depends on the past $m$ states. Although the probability structure of a higher-order Markov chain breaks the fundamental Markov assumption, any higher-order Markov chain can be reduced to a first-order, or standard, Markov chain by taking a Cartesian product of its state space. Consider, for example, a second-order $n$-state Markov chain $S$. Its transition probabilities are $\\ensuremath{\\cel{P}}_{ijk}=\\Pr(S_{t+1}=i\\mid S_t=j, S_{t-1}=k)$. We will represent these probabilities as a tensor $\\cmP$. The stationary distribution equation for the resulting first-order Markov chain satisfies\n\\[\n\\sum_{k} \\ensuremath{\\cel{P}}_{ijk} X_{jk} = X_{ij} ,\n\\]\nwhere $X_{jk}$ denotes the stationary probability on the product space.  Here, we have induced an $n^2 \\times n^2$ eigenvector problem to compute such a stationary distribution. For such first-order Markov chains, Perron-Frobenius theory~\\cite{Perron-1907-theorem,Frobenius-1908-theorem,varga1962-book} governs the conditions when the stationary distribution exists. However, in practice for a $100,000 \\times 100,000 \\times 100,000$ tensor, we need to store $10,000,000,000$ entries in $\\mX = [ X_{ij} ]$. This makes it infeasible to work with large, sparse problems. \n\n\\subsection{Li and Ng's approximation}\n\\label{sec:li}\nAs a computationally tractable alternative to working with a first-order chain on the product state-space, \\citet{Li-2013-tensor-markov-chain} define a new type of stationary distribution for a higher-order Markov chain. Again, we describe it for a second-order chain for simplicity. For each term $X_{ij}$ in the stationary distribution they substitute a product $x_i x_j$, and thus for the matrix $\\mX$ they substitute a rank-1 approximation $\\mX = \\vx \\vx^{T}$  where $\\sum_i x_i = 1$. Making this substitution and then summing over $j$ yields an eigenvalue expression called an $l^2$-eigenvalue by \\citet{Lim-2005-eigenvalues} and called a $Z$-eigenvalue by \\citet{Qi-2005-Z-eigenvalues} (one particular type of tensor eigenvalue problem) for $\\vx$:\n\\[\n\\sum_j\\Big(\\sum_k \\elm{P}_{ijk}x_j x_k \\Big) = \\sum_j x_i x_j = x_i \\quad \\Leftrightarrow \\quad \\tensor{P} \\vx^2 = \\vx,\n\\]\nwhere we've used the increasingly common notational convention: \n\\[ \\textstyle [\\tensor{P} \\vx^2 ]_i = \\sum_{jk} \\elm{P}_{ijk} x_j x_k \\] from \\citet{Qi-2005-Z-eigenvalues}.\nAll of these results extend beyond second-order chains, in a relatively straightforward manner. Li and Ng present a series of theorems that govern existence and uniqueness for such stationary distributions that we revisit later. \n\n\\section{Higher-order PageRank} \\label{sec:hopr}\nRecall the PageRank random surfer. With probability $\\alpha$, the surfer transitions according to the Markov chain; and with probability $1-\\alpha$, the surfer transitions according to the fixed distribution $\\vv$. We define a higher-order PageRank by modeling a random surfer on a higher-order chain. With probability $\\alpha$, the surfer transitions according to the higher-order chain; and with probability $1-\\alpha$, the surfer teleports according to the distribution $\\vv$. That is, if $\\cmP$ is the transition tensor of the higher-order Markov chain, then the higher-order PageRank chain has a transition tensor $\\cmM$ where   \n\\[ \\ensuremath{\\cel{M}}(i,j,\\dots,\\ell,k) = \\alpha \\ensuremath{\\cel{P}}(i,j,\\dots,\\ell,k) + (1-\\alpha) v_i. \\]\nRecall that any higher-order Markov chain can be reduced to a first-order chain by taking a Cartesian product of the state space. We call this the reduced form a higher-order Markov chain and in the following example, we explore the reduced form of a second-order PageRank modification.\n\n\\begin{example} \\label{ex:simple-3}\nConsider the following transition probabilities: \n\\[\n\\ensuremath{\\cel{P}}(\\cdot , \\cdot ,1)=\\begin{bmatrix}\n\t\t\t\t0 & \\tfrac{1}{2} & 0\\\\\n\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t1 & \\tfrac{1}{2} & 1\n\t\t\t\\end{bmatrix};\\quad\n\\ensuremath{\\cel{P}}( \\cdot , \\cdot ,2)=\\begin{bmatrix}\n\t\t\t\t\\frac{1}{2} & 0 & 1 \\\\\n\t\t\t\t0 & \\frac{1}{2} & 0 \\\\\n\t\t\t\t\\frac{1}{2} & \\frac{1}{2} & 0\n\t\t\t\\end{bmatrix};\\quad\n\\ensuremath{\\cel{P}}(\\cdot , \\cdot ,3)=\\begin{bmatrix}\n\t\t\t\t\\frac{1}{2} & \\frac{1}{2} & 0 \\\\\n\t\t\t\t0 & \\frac{1}{2} & 0 \\\\\n\t\t\t\t\\frac{1}{2} & 0 & 1\n\t\t\t\\end{bmatrix}.\n\\]\nFigure~\\ref{fig:hyper_tran} shows the state-space transition diagram for the reduced form of the chain before and after its PageRank modification.\n\\end{example}\n\nWe define a higher-order PageRank \\emph{tensor} as the stationary distribution of the reduced Markov chain, organized so that $\\ensuremath{\\cel{X}}(i,j,\\dots,\\ell)$ is the stationary probability associated with the sequence of states $\\ell \\to \\cdots \\to j \\to i$.\n\n\\begin{definition}[Higher-order PageRank]\nLet $\\cmP$ be an order-$m$ transition tensor representing an $(m-1)$\\textsuperscript{th} order Markov chain,\n$\\alpha$ be a probability less than $1$, and $\\vv$ be a stochastic vector. Then the higher-order PageRank tensor $\\cmX$ is the order-$(m-1)$, $n$-dimensional tensor that solves the linear system: \n\\[ \\ensuremath{\\cel{X}}(i,j,\\dots,\\ell) = \\alpha \\sum_{k} \\ensuremath{\\cel{P}}(i,j,\\dots,\\ell,k) \\ensuremath{\\cel{X}}(j, \\dots, \\ell, k) + (1-\\alpha) v_i \\sum_{k} \\ensuremath{\\cel{X}}(j,\\dots,\\ell,k). \\]\n\\end{definition}\n\nFor the second-order case from Example~\\ref{ex:simple-3}, we now write this linear system in a more traditional matrix form in order to make a few observations about its structure. Let $\\mX$ be the PageRank tensor (or matrix, in this case). We have:\n\\begin{equation}\n\\label{eq:1-step chain}\n\\text{vec}(\\mX) = \\left[\\alpha \\mP + (1-\\alpha)\\mV \\right] \\text{vec}(\\mX),\n\\end{equation}\nwhere $\\mP, \\mV \\in \\RR^{n^2 \\times n^2}$, and $\\mV =  \\ve^{T} \\otimes \\eye \\otimes \\vv$. In this setup, the matrix $\\mP$ is sparse and highly structured:\n\\[ \\mP = \\bmat{ \n0 & 0   & 0 & 1\/2 & 0   & 0 & 1\/2 &   0 & 0\\\\\n0 & 0   & 0 & 0   & 0   & 0 & 0   &   0 & 0\\\\\n1 & 0   & 0 & 1\/2 & 0   & 0 & 1\/2 &   0 & 0\\\\ \n0 & 1\/2 & 0 & 0   & 0   & 0 & 0   & 1\/2 & 0\\\\\n0 & 0   & 0 & 0   & 1\/2 & 0 & 0   & 1\/2 & 0\\\\\n0 & 1\/2 & 0 & 0   & 1\/2 & 0 & 0   &   0 & 0\\\\\n0 & 0   & 0 & 0   & 0   & 1 & 0   &   0 & 0\\\\\n0 & 0   & 0 & 0   & 0   & 0 & 0   &   0 & 0\\\\\n0 & 0   & 1 & 0   & 0   & 0 & 0   &   0 & 1\\\\\n}. \\] \nWhen $\\alpha = 0.85$ and $\\vv = (1\/3) \\ve$, the higher-order PageRank matrix is: \n\\[ \\mX = \\bmat{     0.0411 &   0.0236  &  0.0586 \\\\\n                    0.0062 &   0.0365  &  0.0397 \\\\\n                    0.0761 &   0.0223  &  0.6959 }. \\]\nMore generally, both $\\mP$ and $\\mV$ have the following structure for the second-order case:\n\\[\n\\sbmat{\n\\elm{P}_{111}, v_1 & 0 & \\cdots & 0 & \\elm{P}_{112}, v_1 & 0 & \\cdots & 0 & \\cdots & \\elm{P}_{11n}, v_1 & 0 & \\cdots & 0\\\\\n\\elm{P}_{211}, v_2  & 0 & \\cdots & 0 & \\elm{P}_{212}, v_2 & 0 & \\cdots & 0 & \\cdots & \\elm{P}_{21n}, v_2 & 0 & \\cdots & 0\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots & \\vdots & \\ddots &\\vdots\\\\\n\\elm{P}_{n11}, v_n & 0 & \\cdots & 0 & \\elm{P}_{n12}, v_n & 0 & \\cdots & 0 & \\cdots & \\elm{P}_{n1n}, v_n & 0 & \\cdots & 0\\\\\n\\vdots & \\vdots &  \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots & \\vdots & \\ddots &\\vdots\\\\\n0 & \\cdots & 0 & \\elm{P}_{1n1}, v_1 & 0 & \\cdots & 0 & \\elm{P}_{1n2}, v_1 & \\cdots & 0 & \\cdots & 0 & \\elm{P}_{1nn}, v_1\\\\ \n0 & \\cdots & 0 & \\elm{P}_{2n1}, v_2 & 0 & \\cdots & 0 & \\elm{P}_{2n2}, v_2 & \\cdots & 0 & \\cdots & 0 & \\elm{P}_{2nn}, v_2\\\\ \n\\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots &\\vdots\\\\\n0 & \\cdots & 0 & \\elm{P}_{nn1}, v_n & 0 & \\cdots & 0 & \\elm{P}_{nn2}, v_n & \\cdots & 0 & \\cdots & 0 & \\elm{P}_{nnn}, v_n\\\\\n}. \n\\]\n\n\\begin{figure}[!ht]\n\\centering        \n\\subfigure[The higher-order Markov chain]{\\includegraphics[width=0.45\\linewidth]{hyper_tran_no_weight}}\n\\subfigure[The higher-order PageRank chain]{\\includegraphics[width=0.45\\linewidth]{hyper_tran_pagerank}}\n\\caption{The state space transitions for a higher-order Markov chain on the product-space and the PageRank modification of that same chain with new transitions indicated in red. The transitions for both chains must satisfy $\\langle j,k \\rangle \\to \\langle i,j \\rangle$. Note that, unlikely the PageRank modification of a first-order Markov chain, the reduced form of the higher-order PageRank chain does not have a complete set of transitions. For instance, there is no transition between $\\langle 2,3 \\rangle$ and $\\langle 1,3 \\rangle$. }\n\\label{fig:hyper_tran}\n\\end{figure}\n\n\nIn the remainder of this section, we wish to show the relationship between the reduced form of a higher-order PageRank chain and the definition of the PageRank problem (Definition~\\ref{def:pr}). This is not as trivial as it may seem! For instance, in the second-order case, equation~\\eqref{eq:1-step chain} is not of the correct form for Definition~\\ref{def:pr}. But a slight bit of massaging produces the equivalence. \n\nConsider the vectorized equation for the stationary distribution matrix for the second-order case (from equation~\\ref{eq:1-step chain}) as: \n\\[ \\tvec(\\mX) = \\underbrace{[\\alpha \\mP + (1-\\alpha) \\mV]}_{\\mM} \\tvec(\\mX). \\] \nOur goal is to derive a PageRank problem in the sense of Definition~\\ref{def:pr} to find $\\tvec(\\mX)$. As it turns out, $\\mM^2$ will give us this PageRank problem. The idea is this: in the first-order PageRank problem we lose all history after a single teleportation step by construction. In this second-order PageRank problem, we keep around one more state of history, hence, two steps of the second-order chain are required to see the effect of teleportation as in the standard PageRank problem. Formally, the matrix $\\mM^2$ can be written in terms of matrix $\\mP$ and $\\mV$, i.e., \n\\[\n\\mM^2 = \\alpha^2 \\mP^2 + \\alpha(1-\\alpha)\\mP\\mV + \\alpha(1-\\alpha)\\mV\\mP+(1-\\alpha)^2\\mV^2.\n\\]\nWe now show that $\\mV^2 = (\\vv \\kron \\vv)(\\ve^{T} \\kron \\ve^{T})$ by exploiting two properties of the Kronecker product: $(\\mA \\kron \\mB)(\\mC \\kron \\mD) = (\\mA \\mC) \\kron (\\mB \\mD)$ and $\\va^{T} \\kron \\vb = \\vb \\va^{T}$. Note that: \n\\[ \\mV^2 = (\\ve^{T} \\kron \\mI \\kron \\vv) (\\ve^{T} \\kron \\mI \\kron \\vv) = [ \\ve^{T} (\\ve^{T} \\kron \\mI) ] \\kron [ (\\mI \\kron \\vv) \\vv] = (\\ve^{T} \\kron \\ve^{T}) \\kron (\\vv \\kron \\vv). \\]\nThis enables us to write a PageRank equation for $\\tvec(\\mX)$: \n\\[ \\begin{aligned}\n \\tvec(\\mX) & = \\mM^2 \\tvec(\\mX) \\\\\n            & = \\alpha (2-\\alpha) \\underbrace{\\left[ \\tfrac{\\alpha}{2-\\alpha} \\mP^2 + \\tfrac{1-\\alpha}{2-\\alpha} (\\mP \\mV + \\mV \\mP) \\right] }_{\\mP_{\\text{pr}}} \\tvec(\\mX) + (1 - 2 \\alpha + \\alpha^2) \\vv \\kron \\vv, \\end{aligned}\\]\nwhere we used the normalization $\\ve^{T} \\tvec(\\mX) = 1$. Thus we conclude:\n\\begin{lemma} Consider a second-order PageRank problem $\\alpha, \\cmP, \\vv$. Let $\\mP$ be the matrix for the reduced form of $\\cmP$. Let $\\mM = \\alpha \\mP + (1-\\alpha) \\mV$ be the transition matrix for the vector representation of the stationary distribution $\\mX$. This stationary distribution is the PageRank vector of a PageRank problem $(2\\alpha-\\alpha^2, \\mP_{pr}, \\vv\\kron \\vv)$ in the sense of Definition~\\ref{def:pr} with\n\\[\n\\mP_{\\text{pr}} =\\frac{\\alpha}{2-\\alpha}\\mP^2 + \\frac{1-\\alpha}{2-\\alpha}\\mP\\mV + \\frac{1-\\alpha}{2-\\alpha}\\mV\\mP.\n\\]\n\\end{lemma}\n\nAnd we generalize: \n\n\\begin{theorem} \nConsider a higher-order PageRank problem $\\alpha, \\cmP, \\vv$ where $\\cmP$ is an order-$m$ tensor.  Let $\\mP$ be the matrix for the reduced form of $\\cmP$. Let $\\mM = \\alpha \\mP + (1-\\alpha) \\mV$ be the transition matrix for the vector representation of the order-$(m-1)$, $n$-dimensional stationary distribution tensor $\\cmX$. This stationary distribution is equal to the PageRank vector of the PageRank problem \n\\[ (1-(1-\\alpha)^{m-1}, \\mP_{\\text{pr}}, \\fullkron{\\vv}{m-1}), \\text{ where }\n\\mP_{\\text{pr}} = \\frac{\\mM^{m-1}-(1-\\alpha)^{m-1}\\mV^{m-1}}{1-(1-\\alpha)^{m-1}}. \\]\n\\end{theorem}\n\\begin{proof}\nWe extend the previous proof as follows. The matrix $\\mM$ is nonnegative and has only a single recurrent class of all nodes consisting of all nodes in the \\emph{reach} of the set of non-zero entries in $v_i$. Thus, the stationary distribution is unique. We need to look at the $m-1$ step transition matrix to find the PageRank problem. Consider $\\vec{\\cmX}$ as the stationary distribution eigenvector of the $m-1$ step chain: \n\\[ \\tvec(\\cmX) = \\mM \\tvec(\\cmX) = \\mM^{m-1} \\tvec(\\cmX). \\]\nThe matrix $\\mM^{m-1}$ can be written in terms of matrix $\\mP$ and $\\mV$, i.e.,\n\\[\n\\mM^{m-1} = \\bigl( (\\alpha \\mP + (1-\\alpha)\\mV)^{m-1} - (1-\\alpha)^{m-1}\\mV^{m-1} \\bigr) + (1-\\alpha)^{m-1}\\mV^{m-1}.\n\\]\nThe matrix $\\mV$ has the structure \n\\[ \\mV = \\ve^{T} \\kron (\\fullkron{\\mI}{m-2})\\kron \\vv. \\]\nWe now expand $\\mV^{m-1}$ using the property the property of Kronecker products $(\\mA \\kron \\mB)(\\mC \\kron \\mD) = (\\mA \\mC) \\kron (\\mB \\mD)$, repeatedly: \n\\[ \\begin{aligned}\n\\mV^{m-1} & = \\Bigl[ \\ve^{T} \\kron (\\fullkron{\\mI}{m-2})\\kron \\vv \\Bigr] \\cdots \\Bigl[ \\ve^{T} \\kron (\\fullkron{\\mI}{m-2})\\kron \\vv \\Bigr] \\\\\n& = \n\t\\Bigl[ \\ve^{T}(\\ve^{T} \\kron \\mI)(\\ve^{T} \\kron \\mI \\kron \\mI) \\cdots (\\ve^{T} \\kron \\fullkron{\\mI}{m-2} ) \\Bigr] \\kron  \\\\\n\t& \\qquad \\Bigl[ (\\fullkron{\\mI}{m-2} \\vv ) (\\mI \\kron \\mI \\kron \\vv) (\\mI \\kron \\vv) \\vv \\Bigr]\\\\\n& = (\\fullkron{\\ve^{T}}{m-1}) \\kron (\\fullkron{\\vv}{m-1}) \\\\\n &  = (\\fullkron{\\vv}{m-1})(\\fullkron{\\ve^{T}}{m-1}).\n\\end{aligned}\n\\]\nAt this point, we are essentially done as we have shown that the \\emph{stochastic} $\\mM^{m-1}$ has the form $\\mM^{m-1} = \\mZ + (1-\\alpha)^{m-1} (\\vv \\kron \\cdots \\kron \\vv)(\\ve^{T} \\kron \\cdots \\kron \\ve^{T}).$ The statements in the theorem follow from splitting $\\mM^{m-1} = \\alpha_{\\text{pr}} \\mP_{\\text{pr}} + (1-\\alpha_{\\text{pr}}) (\\vv \\kron \\cdots \\kron \\vv)(\\ve^{T} \\kron \\cdots \\kron \\ve^{T})$, that is,\n\\[ \\alpha_{\\text{pr}} = 1- (1-\\alpha)^{m-1} \\qquad \\mP_{\\text{pr}} = \\frac{1}{\\alpha_{\\text{pr}}} \\left( \\mM^{m-1} - (1-\\alpha) \\mV^{m-1} \\right). \\]\nThe matrix $\\mP_{\\text{pr}}$ is stochastic because the final term in the expansion of $\\mM^{m-1}$ is $(1-\\alpha) \\mV^{m-1}$, thus, the remainder is a nonnegative matrix with column sums equal to a single constant less than $1$.\n\\end{proof}\n\n\n\\begin{corollary} \\label{thm:hopr}\nThe higher-order PageRank stationary distribution tensor $\\cmX$ always exists and is unique. Also, the standard PageRank iteration will result in a $1$-norm error of $2\\bigl(1-(1-\\alpha)^{m-1}\\bigr)^k$ after $(m-1)k$ iterations.\n\\end{corollary}\n\nHence, we retain all of the attractive features of PageRank in the higher-order PageRank problem. The uniqueness and convergence results in this section are not overly surprising and simply clarify the relationship between the higher-order Markov chain and its PageRank modification.\n\n\\section{Multilinear PageRank}\n\\label{sec:tensor-pr}\n\nThe tensor product structure in the state-space and the higher-order stationary distribution make the straightforward approaches of the previous section difficult to scale to large problems, such as those encountered in modern bioinformatics and social network analysis applications. The scalability limit is the memory required. Consider an $n$-state, second-order PageRank chain: $(\\alpha, \\cmP, \\vv)$. It requires $O(n^2)$ memory to represent the stationary distribution, which quickly grows infeasible as $n$ scales. To overcome this scalability limitation, we consider the Li and Ng approximation to the stationary distribution with the additional assumption:\n\n\\textsc{Assumption}.\n\\emph{There exists a method to compute $\\cmP \\vx^2$ that works in time proportional to the memory used for to represent $\\cmP$.}\n\nThis assumption mirrors the fast matrix-vector product operator assumption in iterative methods for linear systems. Although here it is critical because there must be at least $n^2$ non-zeros in any second-order stochastic tensor $\\cmP$. If we could afford that storage then the higher-order techniques from the previous section would apply and we would be under the scalability limit. We discuss how to create such fast operators from sparse datasets in Section~\\ref{sec:fast-operators}.\n\nThe Li and Ng approximation to the stationary distribution of a second-order Markov chain replaces the stationary distribution with a symmetric rank-1 factorization: $\\mX = \\vx \\vx^{T}$ where $\\sum_i x_i = 1$. For a second-order PageRank chain, this transformation yields an implicit expression for $\\vx$: \n\\begin{equation} \\label{eq:tensorpr1}\n\\vx = \\alpha \\cmP \\vx^2 + (1-\\alpha) \\vv. \n\\end{equation} \nWe prefer to write this equation in terms of the Kronecker product structure of the tensor flattening along the first index. Let $\\mR := \\cmP_{(1)} = \\flat_1(\\cmP)$ be the $n$-by-$n^2$, stochastic flattening (see \\citealt[Section 12.4.5]{Golub-2013-book} for more on \\emph{flattenings} or \\emph{unfoldings} of a tensor, and \\citealt{Draisma-2014-bounded-rank} for the $\\flat$ notation) along the first index: \n\\[\n\\mR = \\left[\n\\begin{array}{c c c | c c c | c | c c c}\n\\elm{P}_{111} & \\cdots & \\elm{P}_{1n1} & \\elm{P}_{112} & \\cdots & \\elm{P}_{1n2}  & \\cdots & \\elm{P}_{11n} & \\cdots & \\elm{P}_{1nn}\\\\\n\\elm{P}_{211} & \\cdots & \\elm{P}_{2n1} & \\elm{P}_{212} & \\cdots & \\elm{P}_{2n2} & \\cdots & \\elm{P}_{21n} & \\cdots & \\elm{P}_{2nn}\\\\\n\\vdots & \\ddots & \\vdots &  \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\elm{P}_{n11} & \\cdots & \\elm{P}_{nn1}  & \\elm{P}_{n12} & \\cdots & \\elm{P}_{nn2} & \\cdots & \\elm{P}_{n1n} & \\cdots & \\elm{P}_{nnn}\\\\\n\\end{array} \\right].\n\\]\nThen equation~\\ref{eq:tensorpr1} is: \n\\[ \\vx = \\alpha \\mR (\\vx \\kron \\vx) + (1-\\alpha) \\vv. \\]\nConsider the tensor $\\cmP$ from Example~\\ref{ex:simple-3}. The multilinear PageRank vector for this case with $\\alpha = 0.85$ and $\\vv = (1\/3) \\ve$ is: \n\\[ \\vx = \\bmat{     0.1934 \\\\  0.0761 \\\\  0.7305 } \\]\nWe generalize this second-order case to the order-$m$ case in the following definition of the multilinear PageRank problem. \n\n\\begin{definition}[Multilinear PageRank]\n Let $\\cmP$ be an order-$m$ tensor representing an $(m-1)$\\textsuperscript{th} order Markov chain, $\\alpha$ be a probability less than $1$, and $\\vv$ be a stochastic vector. Then the multilinear PageRank vector is a nonnegative, stochastic solution of the following system of polynomial equations: \n \\begin{equation} \\label{eq:tensor-pr}\n  \\vx = \\alpha \\cmP \\vx^{(m-1)} + (1-\\alpha) \\vv \n\n\\quad \\text{ or equivalently } \\quad\n\n   \\vx = \\alpha \\mR (\\fullkron{\\vx}{m-1}) + (1-\\alpha) \\vv\n \\end{equation}\n where $\\mR := \\cmP_{(1)} = \\flat_1(\\cmP)$ is an $n$-by-$n^{(m-1)}$ stochastic matrix of the flattened tensor along the first index.\n\\end{definition}\n\nWe chose the name \\emph{multilinear PageRank} instead of the alternative \\emph{tensor PageRank} to emphasize the multilinear structure in the system of polynomial equations rather than the tensor structure of $\\cmP$. Also, because the tensor structure of $\\cmP$ is shared with the higher-order PageRank vector, which could have then also been called a \\emph{tensor PageRank}. \n\nA multilinear PageRank vector $\\vx$ always exists because it is a special case of the stationary distribution vector considered by Li and Ng. In order to apply their theory, we consider the equivalent problem: \n\\begin{equation} \\label{eq:tpr-ng}\n \\vx = (\\alpha \\mR + (1-\\alpha) \\vv \\ve^{T}) (\\vx \\kron \\cdots \\kron \\vx) = \\cel{\\mat{\\bar{P}}} \\vx^{m-1}, \n\\end{equation}\nwhere $\\cel{\\mat{\\bar{P}}}$ is the \\emph{stochastic} transition tensor whose flattening along the first index is the matrix $\\alpha \\mR + (1-\\alpha) \\vv \\ve^{T}$. The existence of a stochastic solution vector $\\vx$ is guaranteed by Brouwer's fixed point theorem, and more immediately, by the stationary distributions considered by Li and Ng. The existing bulk of Perron-Frobenius theory for nonnegative tensors~\\cite{Lim-2005-eigenvalues,Chang-2008-perron,Friedland-2013-perron}, unfortunately, is not helpful with existence of uniqueness issues as it applies to problems where $\\normof[2]{\\vx} = 1$ instead of the 1-norm.\n\nAlthough the multilinear PageRank vector always exists, it may not be unique as shown in the following example:\n\\begin{example}\nLet $\\alpha = 0.99$, $\\vv = [0, 1, 0]^{T}$, and \n\\[ \\mR = \\bmat{\n                0 &        0     &    0     &    0      &    0     &   0    & 1\/3   & 1  &  0 \\\\\n                0 &        0     &    0     &    0      &    1     &    0   & 1\/3   & 0  &  1 \\\\\n                1 &         1    &     1    &      1     &     0   &      1 &   1\/3 &   0 &   0 }. \\]\nThen both $\\vx = [0, 1, 0]^{T}$ and $\\vx = [0.1890, 0.3663, 0.4447]^{T}$ solve the multilinear PageRank problem.\n\\end{example}\n\n\\subsection{A stochastic process: the spacey random surfer} \\label{sec:process}\nThe PageRank vector is equivalently the stationary distribution of the random surfer stochastic process. The multilinear PageRank equation is the stationary distribution of a stochastic process with a history dependent behavior that we call the \\emph{spacey random surfer}. For simplicity, we describe this for the case of a second-order problem. Let $S_t$ represent the stochastic process for the spacey random surfer. The process depends on the probability table for a second-order Markov chain $\\cmP$. Our motivation is that the spacey surfer would like to transition as the higher-order PageRank Markov chain, $\\Pr (S_{t+1} = i \\mid S_t = j, S_{t-1} = k ) = \\alpha \\ensuremath{\\cel{P}}_{ijk} + (1-\\alpha) v_i$,  however, on arriving at $S_t = j$, the surfer \\emph{spaces out} and \\emph{forgets} that $S_{t-1} = k$. Instead of using the true history state, the spacey random surfer decides to \\emph{guess} that they came from a state they've visited frequently. Let $Y_t$ be a random state that the surfer visited in the past, chosen according to the frequency of visits to that state. (Hence, $Y_t = k$ is more likely if the surfer visited state $k$ frequently in the past.) The spacey random surfer then transitions as: \n\\[ \\Pr(S_{t+1} = i \\mid S_t = j, Y_{t} = k ) = \\alpha \\ensuremath{\\cel{P}}_{ijk} + (1-\\alpha) v_i. \\]\n\nLet us now state the resulting process slightly more formally. Let $\\mathcal{F}_t$ be the natural filtration on the history of the process $S_1, \\dots, S_t$. Then\n\\[ \\Pr (Y_t = k \\mid \\mathcal{F}_t ) = \\frac{1}{t+n} \\left(1 + \\sum_{r=1}^{t} \\Indof{ S_r = k } \\right), \\]\nwhere $\\Indof{\\cdot}$ is the indicator event. In this definition, note that we assume that there is an initial probability of $1\/n$ of $Y_t$ taking any state. For instance, if $n=10$ and $S_1 = 5, S_2 = 6, S_3 = 5$ and $t = 3$, then $Y_t$ is a random variable that takes value $6$ with probability $2\/13$ and value $5$ probability $3\/13$. The stochastic process progresses as: \n\\[ \\Pr (S_{t+1} = i \\mid \\mathcal{F}_t ) = \\alpha \\sum_{k=1}^n \\ensuremath{\\cel{P}}(i,S_{t},k) \\frac{1 + \\sum_{r=1}^{t} \\Indof{ S_r = k }}{t+n} + (1-\\alpha) v_i. \\]\nThis stochastic process is a new type of vertex reinforced random walk~\\cite{Pemantle-1992-vertex-reinforced}. \n\nWe present the following heuristic justification for the equivalence of this process with the multilinear PageRank vector. In our subsequent manuscript~\\citet{Gleich-preprint-srs}, we use results from \\citet{Benaim-1997-vrrw} to make this equivalence precise and also, to study the process in more depth. Suppose the process has run for a long time $t \\gg 1$. Let $\\vy$ be the probability distribution of selecting any state as $Y_t$. The vector $\\vy$ changes slowly when $t$ is large. For some time in the future, we can approximate the transitions as a first-order Markov chain: \n\\[ \\Pr(S_{t+1} = i \\mid S_t = j) \\approx \\alpha \\ensuremath{\\cel{P}}_{i,j,k} y_k + (1-\\alpha) v_i. \\] \nLet $\\mR_k = \\ensuremath{\\cel{P}}(:,:,k)$ be a slice of the probability table, then the Markov transition matrix is: \n\\[ \\alpha \\sum_{k=1}^n \\mR_k y_k + (1-\\alpha) \\vv = \\alpha \\mR (\\vy \\kron \\mI) + (1-\\alpha) \\vv \\ve^{T}. \\]\nThe resulting stationary distribution is a vector $\\vx$ where: \n\\[ \\vx = \\alpha \\mR( \\vy \\kron \\vx) + (1-\\alpha) \\vv . \\]\nIf $\\vy = \\vx$, then the distribution of $\\vy$ will not change in the future, whereas if $\\vy \\not= \\vx$, then the distribution of $\\vy$ \\emph{must change} in the future. Hence, we must have $\\vx = \\vy$ at stationarity and any stationary distribution of the spacey random surfer must be a solution of the multilinear PageRank problem.\n\n\\subsection{Sufficient conditions for uniqueness}\nIn this section, we provide a sufficient condition for the multilinear PageRank vector to be unique. Our original conjecture was that this vector would be unique when $\\alpha < 1$, which mirrors the case of the standard and higher-order PageRank vectors; however, we have already seen an example where this was false. Throughout this section, we shall derive and prove the following result:\n\\begin{theorem} \\label{thm:tpr-unique}\n Let $\\cmP$ be an order-$m$ stochastic tensor, $\\vv$ be a nonnegative vector. Then the multilinear PageRank equation \n \\[ \\vx = \\alpha \\cmP \\vx^{(m-1)} + (1-\\alpha) \\vv \\]\n has a unique solution when $\\alpha < \\frac{1}{m-1}$.\n\\end{theorem}\n\nTo prove this statement, we first prove a useful lemma about the norm of the difference of the Kronecker products between two stochastic vectors with respect to the difference of each part. We suspect this result is known, but were unable to find an existing reference.\n\\begin{lemma}\n\\label{kronecker_product_norm}\n\\label{lem:stochastic-diff-2}\nLet $\\va, \\vb, \\vc,$ and $\\vd$ be stochastic vectors where $\\va$ and $\\vc$ have the same size. The 1-norm of the difference of their Kronecker products satisfies the following inequality,\n\\[ \\normof[1]{ \\va \\kron \\vb - \\vc \\kron \\vd } \\le \\normof[1]{\\va - \\vc} + \\normof[1]{\\vb - \\vd}. \\]\n\\end{lemma}\n\\begin{proof}\nThis proof is purely algebraic and begins by observing:\n\\[ \\va \\kron \\vb - \\vc \\kron \\vd \n\t= \\frac{1}{2} (\\va - \\vc) \\kron (\\vb + \\vd) + \\frac{1}{2} (\\va + \\vc) \\kron (\\vb - \\vd). \\]\nIf we separate the bound into pieces we must bound terms such as $\\normof[1]{ (\\va - \\vc) \\kron (\\vb + \\vd) }$. But by using the stochastic property of the vectors, this term equals $\\sum_{ij} (b_j + d_j) |a_i - c_i| = 2\\normof[1]{\\va - \\vc}.$\n\\end{proof}\n\nThis result is essentially tight. Let us consider two stochastic vectors of 2 dimensions, $\\vx=[x_1, 1-x_1]^{T}$ and $\\vy=[y_1, 1-y_1]^{T}$, where $x_1 \\neq y_1$.\nThen, \n\\[\n\\frac{\\normof[1]{\\vx \\kron \\vx - \\vy \\otimes \\vy}}{\\normof[1]{\\vx - \\vy_1}} = \\frac{1}{2}|x_1 + y_1| + |1-(x_1+y_1)| + \\frac{1}{2}|2 - (x_1+y_1)|.\n\\]\nThe ratio of $\\normof[1]{\\vx \\kron \\vx - \\vy \\kron \\vy} \/ \\normof[1]{\\vx - \\vy}$ approaches to 2 as $x_1+y_1 \\rightarrow 0$ or $x_1 + y_1 \\rightarrow 2$. However, this bound cannot be achieved.\n\nThe conclusion of Lemma \\ref{kronecker_product_norm} can be easily extended to the case where there are \nmultiple Kronecker products between vectors.\n\\begin{lemma}\n\\label{lem:stochastic-diff}\nFor stochastic vectors $\\vx_1, \\dots, \\vx_m$ and $\\vy_1, \\dots, \\vy_m$ where the size of $\\vx_i$ is the same as the size of $\\vy_i$, then  \n\\[ \\normof[1]{  \\vx_1 \\kron \\cdots \\kron \\vx_m - \\vy_1 \\kron \\cdots \\kron \\vy_m } \\le \\sum_{i} \\normof[1]{\\vx_i - \\vy_i}. \\]\n\\end{lemma}\n\\begin{proof}\nLet us consider the case of $m=3$. Let $\\va = \\vx_1 \\kron \\vx_2$, $\\vc = \\vy_1 \\kron \\vy_2$, $\\vb = \\vx_3$ and $\\vd = \\vy_3$. Then \n\\[ \\normof[1]{ \\vx_1 \\kron \\vx_2 \\kron \\vx_3 - \\vy_1 \\kron \\vy_2 \\kron \\vy_3} = \\normof[1]{ \\va \\kron \\vb - \\vc \\kron \\vd} \\le \\normof[1]{\\va - \\vc} + \\normof[1]{\\vx_3 - \\vy_3} \\]\nby using Lemma~\\ref{lem:stochastic-diff-2}. But by recurring on $\\va - \\vc$, we complete the proof for $m=3$. It is straightforward to apply this argument inductively for $m > 3$.\n\\end{proof} \n\nThis result makes it easy to show uniqueness of the multilinear PageRank vectors:\n\n\\begin{lemma}\nThe multilinear PageRank equation has the unique solution when $\\alpha < 1\/2$ for third order tensors.\n\\end{lemma}\n\\begin{proof}\nAssume there are two distinct solutions to the multilinear PageRank equation,\n\\[\n\\begin{aligned}\n\\vx &= \\alpha \\mR(\\vx \\otimes \\vx) + (1-\\alpha)\\vv\\\\\n\\vy &= \\alpha \\mR(\\vy \\otimes \\vy) + (1-\\alpha)\\vv \\\\\n\\vx - \\vy &= \\alpha \\mR (\\vx \\otimes \\vx - \\vy \\otimes \\vy).\n\\end{aligned}\n\\]\nWe simply apply Lemma~\\ref{lem:stochastic-diff-2}:\n\\[\n\\normof[1]{\\vx - \\vy} = \\normof[1]{\\alpha \\mR (\\vx \\otimes \\vx - \\vy \\otimes \\vy)} \\le 2\\alpha\\normof[1]{\\mR} \\normof[1]{\\vx-\\vy} < \\normof[1]{\\vx-\\vy},\n\\]\nwhich is a contradiction (recall that $\\mR$ is stochastic). Thus, the multilinear PageRank equation has the unique solution when $\\alpha < 1\/2$.\n\\end{proof} \\bigskip\n\nThe proof of the general result in Theorem~\\ref{thm:tpr-unique} is identical, except that it uses the general bound  Lemma~\\ref{lem:stochastic-diff} on the order-$m$ problem.\n\n\\subsection{Uniqueness via Li and Ng's results}\n\nLi and Ng's recent paper~\\cite{Li-2013-tensor-markov-chain} tackled the same uniqueness question for the general problem: \\[ \\cmP \\vx^{m-1} = \\vx. \\]  We can also write our problem in this form as in equation~\\ref{eq:tpr-ng} and apply their theory. In the case of a third-order problem, or $m=3$, they define a quantity to determine uniqueness:\n\\[ \\bigbeta. \\] \nFor any tensor where $\\beta > 1$, the vector $\\vx$ that solves \\[ \\cmP \\vx^2 = \\vx \\] is unique. In Appendix~\\ref{sec:pagerank-beta} to this paper, we show that $\\beta > 1$ is a stronger condition that $\\alpha < 1\/2$. We defer this derivation to the appendix as it is slightly tedious and does not result in any new insight into the problem.\n\n\\subsection{PageRank and higher-order PageRank}\n\n\nWe conclude this section by establishing some relationships between multilinear PageRank, higher-order PageRank, and PageRank for a special tensor. In the case when there is no higher-order structure present, then the multilinear PageRank, higher-order PageRank, and PageRank are all equivalent. The precise condition is where $\\mR = \\ve^{T} \\kron \\mQ$ for a stochastic matrix $\\mQ$, which models a higher-order random surfer with behavior that is \\emph{independent} of the last state. Thus, we'd expect that none of our higher-order modifications would change the properties of the stationary distribution. \n\n\\begin{proposition}\nConsider a second-order multilinear PageRank problem with a third-order stochastic tensor where the flattened matrix $\\mR = \\ve^{T} \\kron \\mQ$ has dimension $n \\times n^2$ and where $\\mQ$ is an $n \\times n$ column stochastic matrix. Then for all $0 < \\alpha < 1$ and stochastic vectors $\\vv$, the multilinear PageRank vector is the same as the PageRank vector of $(\\alpha, \\mQ, \\vv)$. Also, the marginal distribution of the higher-order PageRank solution matrix, $\\mX \\ve$, is the same as well.\n\\end{proposition}\n\\begin{proof}\nIf $\\mR = \\ve^{T} \\otimes \\mQ$, then any solution of equation~\\eqref{eq:tensor-pr} is also the unique solution of the standard PageRank equation:\n\\[\n\\vx = \\alpha (\\ve^{T} \\kron \\mQ)(\\vx \\kron \\vx) + (1-\\alpha)\\vv = \\alpha \\mQ \\vx + (1-\\alpha)\\vv.\n\\]\nThus, the two solutions must be the same and the multilinear PageRank problem has a unique solution as well. \nNow consider the solution of the second-order PageRank problem from equation~\\eqref{eq:1-step chain}:\n\\[ \\tvec(\\mX) = [\\alpha \\mP + (1-\\alpha) \\mV ]\\tvec(\\mX). \\]\nNote that $\\mR = (\\ve^{T} \\kron \\mI) \\mP$. Consider the marginal distribution: $\\vy = \\mX \\ve = (\\ve^{T} \\kron \\mI) \\tvec(\\mX)$. The vector $\\vy$ must satisfy: \n\\[ \\vy = (\\ve^{T} \\kron \\mI) \\tvec(\\mX) = (\\ve^{T} \\kron \\mI)  [\\alpha \\mP + (1-\\alpha) \\mV ]\\tvec(\\mX) = \\alpha \\mR \\tvec(\\mX) + (1-\\alpha) \\vv.\\]\nBut $\\mR \\tvec(\\mX) = (\\ve^{T} \\kron \\mQ) \\tvec(\\mX) = \\mQ \\vy$.\n\\end{proof}\n \n\\subsection{Fast operators from sparse data}\n\\label{sec:fast-operators}\nThe last detail we wish to mention is how to build a fast operator $\\cmP \\vx^{m-1}$ when the input tensor is highly sparse. Let $\\cmQ$ be the tensor that models the original \\emph{sparse} data, where $\\cmQ$ has far fewer than $n^{m-1}$ non-zeros and cannot be stochastic. Nevertheless, suppose that $\\cmQ$ has the following property: \n\\[ \\ensuremath{\\cel{Q}}(i, j, \\dots, k) \\ge 0 \\quad \\text{ and } \\quad \\sum_i \\ensuremath{\\cel{Q}}(i, j, \\dots, k) \\le 1 \\text{ for all } j, \\dots, k. \\]\nThis could easily be imposed on a set of nonnegative data in time and memory proportional to the non-zeros of $\\cmQ$ if that were not originally true. To create a fast operator for a fully stochastic problem, we generalize the idea behind the \\emph{dangling indicator} correction of PageRank. (The following derivation is entirely self contained, but the genesis of the idea is identical to the dangling correction in PageRank problems~\\cite{boldi2007-traps}.) Let $\\mS$ be the flattening of $\\cmQ$ along the first index. Let $\\vd^{T} = \\ve^{T} - \\ve^{T} \\mS \\ge 0$, and let $\\vu$ be a stochastic vector that determines what the model should do on a \\emph{dangling case}. Then: \n\\[ \\mR = \\mS + \\vu \\vd^{T} \\]\nis a column stochastic matrix, which we interpret as the flattening of $\\cmP$ along the first index. If $\\vx$ is a stochastic vector, then we can evaluate: \n\\[ \\mR \\vx = \\underbrace{\\mS \\vx}_{\\vz} + \\vu (\\ve^{T} \\vx - \\ve^{T} \\mS \\vx) = \\vz + (1 - \\ve^{T} \\vz) \\vu, \\]\nwhich only involves work proportional to the non-zeros of $\\mS$ or the non-zeros of $\\mQ$.\nThus, given any sparse tensor data, we can create a fully stochastic model.\n\n\n\\section{Algorithms for Multilinear PageRank} \\label{sec:methods}\n\nAt this point, we begin our discussion of algorithms to compute the multilinear PageRank vector. In the following\nsection, we investigate five different methods to compute it. The methods are all inspired by the fixed-point\nnature of the multilinear PageRank solution. They are: \n\\begin{compactenum}\n\\item a fixed-point iteration, as in the power method and Richardson method;\n\\item a shifted fixed-point iteration, as in SS-HOPM~\\cite{Kolda-2011-sshopm};\n\\item a non-linear inner-outer iteration, akin to \\citet{gleich2010-inner-outer};\n\\item an inverse iteration, as in the inverse power method; and\n\\item a Newton iteration.\n\\end{compactenum}\nWe will show that the first four of them converge in the case that $\\alpha < 1\/(m-1)$ for an order-$m$  tensor. For Newton, we show it converges quadratically fast for a third-order tensor when $\\alpha < 1\/2$. We also illustrate a few empirical advantages of each method. \n\n\n\n\\paragraph{The test problems}\nThroughout the following section, the following two problems help illustrate the methods:\n\\[ \\begin{aligned}\n\\mR_1 & = \\left[ \\begin{array}{ccc|ccc|ccc}\n1\/3 & 1\/3 & 1\/3 & 1\/3 & 0 & 0 & 0 & 0 & 0 \\\\\n1\/3 & 1\/3 & 1\/3 & 1\/3 & 0 & 1\/2 & 1 & 0 & 1 \\\\\n1\/3 & 1\/3 & 1\/3 & 1\/3 & 1 & 1\/2 & 0 & 1 & 0 \\\\\n\\end{array} \\right] \\\\\n\\mR_2 & = \\left[ \\begin{array}{cccc|cccc|cccc|cccc}\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0   & 0 & 0 & 0 & 0   & 0 & 1\/2 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1\/2 & 0 & 0 & 0 & 1\/2 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\/2 & 1 & 1 & 0 & 0   & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0   & 0 & 0 & 1 & 1\/2 & 1 & 1\/2 \\\\\n\\end{array} \\right]\n\\end{aligned} \\]\nwith $\\vv = \\ve\/n$. The parameter $\\alpha$ will vary through our experiments, but we are most interested in the regime where $\\alpha > 1\/2$ to understand how the algorithms behave outside of the region where we can prove they converge.\nWe derived these problems by using exhaustive and randomized searches over the\nspace of $2 \\times 2 \\times 2$, $3 \\times 3 \\times 3$, and $4 \\times 4 \\times 4$\nbinary-valued tensors, which we then normalized to be stochastic. Problems $\\mR_1$ and $\\mR_2$ were made from the database of problems we consider from the next section (Section~\\ref{sec:experiments}).\n\nThe residual of a problem and a potential solution $\\vx$ is the $1$-norm:\n\\begin{equation} \\label{eq:residual}\n \\text{residual} = \\normof[1]{ \\alpha \\cmP \\vx^{m-1} + (1-\\alpha) \\vv - \\vx}. \n\\end{equation}\nWe seek methods that cause the residual to drop below $10^{-8}$. For all the examples in this section, we ran the method out to $10,000$ iterations to ensure there was no delayed convergence, although, we only show $1000$ iterations.\n\n\n\n\n\n\n\\subsection{The fixed-point iteration}\n\nThe multilinear PageRank problem seeks a fixed-point of the following non-linear map: \n\\[ f(\\vx) = \\alpha \\cmP \\vx^{m-1} + (1-\\alpha) \\vv. \\]\nWe first show convergence of the iteration implied by this map in the case\nthat $\\alpha < 1\/(m-1)$.\n\\begin{theorem}\nLet $\\cmP$ be an order-$m$ stochastic tensor, let $\\vv$ and $\\vx\\itn{0}$ be stochastic vectors, and let $\\alpha < 1\/(m-1)$. The fixed-point iteration \n\\[ \\vx\\itn{k+1}^{} = \\alpha \\cmP \\vx\\itn{k}^{m-1} + (1-\\alpha) \\vv \\]\nwill converge to the unique solution $\\vx$ of the multilinear PageRank problem~\\eqref{eq:tensor-pr} and also \n\\[ \\normof[1]{\\vx\\itn{k} - \\vx} \\le [\\alpha(m-1)]^k \\normof[1]{\\vx\\itn{0} - \\vx} \\le 2[\\alpha(m-1)]^k. \\]\n\\end{theorem}\n\\begin{proof}\nNote first that this problem has a unique solution $\\vx$ by Theorem~\\ref{thm:tpr-unique}, and also that $\\vx\\itn{k}$ remains stochastic for all iterations. \nThis result is then, essentially, an implication of Lemma~\\ref{lem:stochastic-diff}.\nLet $\\mR$ be the flattening of $\\cmP$ along the first index. \nThen using that Lemma, \n\\[ \\normof[1]{\\vy - \\vx\\itn{k+1}} = \\normof[1]{\\alpha \\mR (\\vy \\kron \\cdots \\kron \\vy - \\vx\\itn{k} \\kron \\cdots \\kron \\vx\\itn{k})} \\le \\alpha (m-1) \\normof[1]{\\vy - \\vx\\itn{k}}. \\]\nThus, we have established a contraction.\n\\end{proof}\n\nLi and Ng treat the same iteration in their paper and they show a more general convergence result that implies our theorem, thus providing a more refined understanding of the convergence of this iteration. However, their result needs a difficult-to-check criteria. In earlier work by~\\cite{Rabinovich-1992-quadratic}, they show that the fixed-point iteration will always converge when a certain symmetry property holds, however, they do not have a rate of convergence. Nevertheless, it is still easy to find PageRank problems that will not converge with this iteration. Figure~\\ref{fig:fp-diverge} shows the result of using this method on $\\mR_1$ with $\\alpha = 0.95$ and $\\alpha = 0.96$. The former converges nicely and the later does not.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\linewidth]{power-R3_1-95}%\n\\includegraphics[width=0.5\\linewidth]{power-R3_1-96}%\n\n\\caption{At left, the components of the iterates of the fixed point iteration for $\\mR_1$ with $\\alpha = 0.95$ show that it converges to a solution. (See the inset residual in the upper right.) At right, the result of the fixed point iteration with $\\alpha=0.96$ illustrates a case that does not converge. }\n\\label{fig:fp-diverge}\n\\end{figure}\n\n\\subsection{The shifted fixed-point iteration}\n\\citet{Kolda-2011-sshopm} noticed a similar phenomenon for the convergence of the symmetric higher-order power method and proposed the \\emph{shifted} symmetric higher-order power method (SS-HOPM) to address these types of oscillations. They were able to show that their iteration always converges monotonically for an appropriate shift value. For the multilinear PageRank problem, we study the iteration given by the equivalent fixed-point: \n\\[ (1+\\gamma) \\vx = \\left[ \\alpha \\cmP \\vx^{m-1} + (1-\\alpha) \\vv \\right] + \\gamma \\vx. \\]\nThe resulting iteration is what we term the \\emph{shifted fixed-point iteration}\n\\[\n \\vx\\itn{k+1}^{} = \\frac{\\alpha}{1+\\gamma} \\cmP \\vx\\itn{k}^{m-1} + \\frac{1-\\alpha}{1+\\gamma} \\vv + \\frac{\\gamma}{1+\\gamma} \\vx\\itn{k}. \n \\]\nIt shares the property that an initial stochastic approximation $\\vx\\itn{0}$ will remain stochastic throughout. \n\n\\begin{theorem}\nLet $\\cmP$ be an order-$m$ stochastic tensor, let $\\vv$ and $\\vx\\itn{0}$ be stochastic vectors, and let $\\alpha < 1\/(m-1)$. The shifted fixed-point iteration \n\\begin{equation} \\label{eq:shifted}\n \\vx\\itn{k+1}^{} = \\frac{\\alpha}{1+\\gamma} \\cmP \\vx\\itn{k}^{m-1} + \\frac{1-\\alpha}{1+\\gamma} \\vv + \\frac{\\gamma}{1+\\gamma} \\vx\\itn{k} \n\\end{equation}\nwill converge to the unique solution $\\vx$ of the multilinear PageRank problem~\\eqref{eq:tensor-pr} and also \n\\[ \\normof[1]{\\vx\\itn{k} - \\vx} \\le \\left( \\frac{\\alpha(m-1) + \\gamma}{1+\\gamma} \\right)^k \\normof[1]{\\vx\\itn{0} - \\vx} \\le 2\\left( \\frac{\\alpha(m-1) + \\gamma}{1+\\gamma} \\right)^k. \\]\n\\end{theorem}\nThe proof of this convergence is, in essence, identical to the previous case and we omit it for brevity. \n\nThis result also suggests that choosing $\\gamma = 0$ is optimal and we should not shift the iteration at all. That is, we should run the fixed-point iteration. This analysis, however, is misleading as illustrated in Figure~\\ref{fig:shifted}. There, we show the iterates from solving $\\mR_1$ with $\\alpha = 0.96$, which did not converge with the fixed-point iteration, but converges nicely with $\\gamma = 1\/2$. However, $\\gamma < (m-2)\/2$ will not guarantee convergence and the same figure shows that $\\mR_2$ with $\\alpha = 0.97$ will not converge. We now present a necessary analysis that shows this method may not converge if $\\gamma < (m-2)\/2$ when $\\alpha > 1\/(m-1)$.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\linewidth]{shifted-R3_1-96}%\n\\includegraphics[width=0.5\\linewidth]{shifted-R4_11-97}%\n\n\\caption{When we use a shift $\\gamma = 1\/2$, then at left, the iterates of the shifted iteration for $\\mR_1$ with $\\alpha = 0.96$ shows that it quickly converges to a solution, whereas this same problem did not converge with the fixed-point method.  At right, the result of the shifted iteration on $\\mR_2$ with $\\alpha=0.97$ again shows a case that does not converge. }\n\\label{fig:shifted}\n\\end{figure}\n\n\\paragraph{On the necessity of shifting} To derive this result, we shall restate the multilinear PageRank problem as the limit point of an ordinary differential equation. There are other ways to derive this result as well, but this one is familiar and relatively straightforward. Consider the ordinary differential equation: \n\\begin{equation} \\label{eq:ode}\n \\frac{d\\vx}{dt} = \\alpha \\cmP \\vx^{m-1} + (1-\\alpha) \\vv - \\vx. \n\\end{equation}\nA forward Euler discretization yields the iteration: \n\\[ \\vx\\itn{k+1}^{} = \\alpha h \\cmP \\vx\\itn{k}^{m-1} + (1-\\alpha) h \\vv + (1-h) \\vx\\itn{k}, \\]\nwhich is identical to the shifted iteration~\\eqref{eq:shifted} with $h = \\frac{1}{1+\\gamma}$. To determine if forward Euler converges, we need to study the Jacobian of the ordinary differential equation. Let $\\mR$ be the flattening of $\\cmP$ along the first index, then the Jacobian of the ODE \\eqref{eq:ode} is: \n\\[ \\mJ(\\vx) = \\alpha \\mR ( \\mI \\kron \\vx \\kron \\cdots \\kron \\vx + \\vx \\kron \\mI \\kron \\vx \\kron \\cdots \\kron \\vx + \\vx \\kron \\cdots \\kron \\vx \\kron \\mI) - \\mI. \\] \nA necessary condition for the forward Euler method to converge is that it is absolutely stable. In this case, we need  $|1-h \\rho(\\mJ)| \\le 1$, where $\\rho$ is the spectral radius of the Jacobian. For all stochastic vectors $\\vx$ generated by iterations of the algorithm, $\\rho(\\mJ(\\vx)) \\le (m-1) \\alpha + 1 \\le m$. Thus, $h \\le 2\/m$ is necessary for a general convergence result when $\\alpha > 1\/(m-1)$ . This, in turn, implies that $\\gamma \\ge (m-2)\/2$. In the case that $\\alpha < 1\/(m-1)$, then the Jacobian already has eigenvalues within the required bounds and no shift is necessary.\n\n\\begin{remark}\nBased on this analysis, we always recommend the shifted iteration with $\\gamma \\ge (m-2)\/2$ for any problem with $\\alpha > 1\/(m-1)$.\n\\end{remark}\n\n\n\n\\subsection{An inner-outer iteration}\n\nWe now develop a non-linear iteration scheme using that uses multilinear PageRank, in the convergent regime, as a subroutine. To derive this method, we use the relationship between multilinear PageRank and the multilinear Markov chain formulation discussed in Section~\\ref{sec:li}. Let $\\mRbar  = \\alpha \\mR + (1-\\alpha) \\vv \\ve^{T}$ then note that this the Markov chain form of the problem is: \n\\[ \\mRbar (\\fullkron{\\vx}{m-1}) = \\vx \\quad \\Leftrightarrow \\quad [ \\alpha \\mR + (1-\\alpha) \\vv \\ve^{T} ] (\\fullkron{\\vx}{m-1}) = \\vx. \\] \nEquivalently, we have: \n\\[ \\frac{\\alpha}{m-1} \\mRbar (\\fullkron{\\vx}{m-1}) + \\left(1-\\frac{\\alpha}{m-1}\\right) \\vx = \\vx. \\]\nFrom here, the nonlinear iteration emerges: \n\\begin{equation} \\vx\\itn{k+1} = \\frac{\\alpha}{m-1} \\mRbar (\\fullkron{\\vx\\itn{k+1}}{m-1}) + \\left(1-\\frac{\\alpha}{m-1}\\right) \\vx\\itn{k}. \\end{equation}\nEach iteration involves solving a multilinear PageRank problem with $\\mRbar, \\alpha\/(m-1),$ and $\\vx\\itn{k}$. Because $\\alpha < 1$, then $\\alpha\/(m-1) < 1\/(m-1)$ and the solution of these subproblems is unique, and thus, the method is well-defined. Not surprisingly, this method also converges when $\\alpha < 1\/(m-1)$.\n\n\\begin{theorem}\nLet $\\cmP$ be an order-$m$ stochastic tensor, let $\\vv$ and $\\vx\\itn{0}$ be stochastic vectors, and let $\\alpha < 1\/(m-1)$. Let $\\mR$ be the flattening of $\\cmP$ along the first index and let $\\mRbar = \\alpha \\mR + (1-\\alpha) \\vv \\ve^{T}$.\n The inner-outer multilinear PageRank iteration\n\\[ \\vx\\itn{k+1} = \\frac{\\alpha}{m-1} \\mRbar (\\fullkron{\\vx\\itn{k+1}}{m-1}) + \\left(1-\\frac{\\alpha}{m-1}\\right) \\vx\\itn{k} \\]\n converges to the unique solution $\\vx$ of the multilinear PageRank problem and also \n \\[ \\normof[1]{\\vx\\itn{k} - \\vx} \\le \\left( \\frac{1-\\alpha\/(m-1)}{1-\\alpha^2} \\right)^{k} \\normof[1]{\\vx\\itn{0} - \\vx} \\le 2 \\left( \\frac{1-\\alpha\/(m-1)}{1-\\alpha^2} \\right)^k. \\]\n\\end{theorem}\n\\begin{proof}\nRecall that this is the regime of $\\alpha$ when the solution is unique. Note that \n\\[ \\begin{aligned}\n\\vx\\itn{k+1} - \\vx \n& = \\frac{\\alpha}{m-1} \\mRbar ( \\fullkron{\\vx\\itn{k+1}}{m-1} - \\fullkron{\\vx}{m-1} ) + \\left(1 - \\frac{\\alpha}{m-1}\\right) (\\vx\\itn{k} - \\vx) \\\\\n& = \\frac{\\alpha^2}{m-1} \\mR ( \\fullkron{\\vx\\itn{k+1}}{m-1} - \\fullkron{\\vx}{m-1} ) + \\left(1 - \\frac{\\alpha}{m-1}\\right) (\\vx\\itn{k} - \\vx). \n\\end{aligned} \\]\nBy using Lemma~\\ref{lem:stochastic-diff}, we can bound the norm of the difference of the $m-1$ term Kronecker products by $(m-1) \\normof[1]{\\vx\\itn{k+1} - \\vx}$. Thus, \n\\[ \\normof[1]{\\vx\\itn{k+1} - \\vx} \\le \\alpha^2 \\normof[1]{\\vx\\itn{k+1} - \\vx} + \\left(1-\\frac{\\alpha}{m-1}\\right) \\normof[1]{\\vx\\itn{k} - \\vx}, \\]\nand the scheme converges linearly with rate $ \\frac{1-\\alpha\/(m-1)}{1-\\alpha^2} < 1$ when $\\alpha < 1\/(m-1)$.\n\\end{proof}\n\nIn comparison with the shifted method, each iteration of the inner-outer method is far more expensive and involves solving a multilinear PageRank method. However, if $\\cmP$ is only available through a fast operator, this may be the only method possible. In Figure~\\ref{fig:innout}, we show that the inner-outer method converges in the case that the shifted method failed to converge. Increasing $\\alpha$ to $0.99$, however, now generates a problem where the inner-outer method will not converge.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\linewidth]{innout-R4_11-97}%\n\\includegraphics[width=0.5\\linewidth]{innout-R4_11-99}%\n\n\\caption{At left, the iterates of the inner-outer iteration for $\\mR_2$ with $\\alpha = 0.97$ shows that it converges to a solution, whereas this same problem did not converge with the shifted method. At right, the result of the shifted iteration on $\\mR_2$ with $\\alpha=0.90$ again shows an example that doesn't converge. }\n\\label{fig:innout}\n\\end{figure}\n\n\\subsection{An inverse iteration}\n\nAnother algorithm we consider is given by our interpretation of the multilinear PageRank solution as a stochastic process.  Observe, for the second-order case, \n\\[ \\alpha \\mR (\\vx \\kron \\vx) = \\tfrac{\\alpha}{2}\\mR (\\vx \\kron \\mI + \\mI \\kron \\vx) = \\alpha \\, [ \\tfrac{1}{2} \\mR (\\vx \\kron \\mI) + \\tfrac{1}{2} \\mR ( \\mI \\kron \\vx) ]. \\]\nBoth matrices \n\\[ \\mR (\\vx \\kron \\mI) \\quad \\text{ and } \\quad \\mR (\\mI \\kron \\vx) \\]\nare stochastic. Let $\\mS(\\vx) = \\tfrac{1}{2} \\mR (\\vx \\kron \\mI) + \\tfrac{1}{2} \\mR ( \\mI \\kron \\vx)$ be stochastic sum of these two matrices. Then the multilinear PageRank vector satisfies: \n\\[ \\vx = \\alpha \\mS(\\vx) \\vx + (1-\\alpha) \\vv. \\]\nThis equation has a subtle interpretation. The multilinear PageRank vector \\emph{is} the PageRank vector of a solution dependent Markov process. The stochastic process presented in Section~\\ref{sec:process} shows this in a slightly different manner. The iteration that arises is a simple fixed-point idea using this interpretation: \n\\[ \\vx\\itn{k+1} = \\alpha \\mS(\\vx\\itn{k}) \\vx\\itn{k+1} + (1-\\alpha) \\vv. \\]\nThus, at each step, we solve a PageRank problem given the current iterate to produce the subsequent vector. For this iteration, we could then leverage a fast PageRank solver if there is a way of computing $\\mS(\\vx\\itn{k})$ effectively or $\\mS(\\vx\\itn{k}) \\vx$ effectively. The method for a general problem is the same, except for the definition of $\\mS$. In general, let\n\\begin{equation} \\label{eq:mlpr-markov}\n \\mS(\\vx) = \\tfrac{1}{m-1} \\mR (\\mI \\kron \\fullkron{\\vx\\itn{k}}{m-2} + \\vx\\itn{k} \\kron \\mI \\kron \\fullkron{\\vx\\itn{k}}{m-3} + \\dots + \\fullkron{\\vx\\itn{k}}{m-2} \\kron \\mI). \n \\end{equation}\nThis iteration is guaranteed to converge in the unique solution regime.\n\n\\begin{theorem}\nLet $\\cmP$ be an order-$m$ stochastic tensor, let $\\vv$ and $\\vx\\itn{0}$ be stochastic vectors, and let $\\alpha < 1\/(m-1)$. Let $\\mS(\\vx\\itn{k})$ be an $n \\times n$ stochastic matrix defined via~\\eqref{eq:mlpr-markov}. The inverse multilinear PageRank iteration \n\\[ \\vx\\itn{k+1} = \\alpha \\mS(\\vx\\itn{k}) \\vx\\itn{k+1} + (1-\\alpha) \\vv \\]\nconverges to the unique solution $\\vx$ of the multilinear PageRank problem and also \n\\[ \\normof[1]{\\vx\\itn{k} - \\vx} \\le  \\left( \\frac{(m-2) \\alpha}{1-\\alpha} \\right)^k \\normof[1]{\\vx\\itn{0} - \\vx} \\le 2 \\left( \\frac{(m-2) \\alpha}{1-\\alpha} \\right)^k. \\]\n\\end{theorem}\n\\begin{proof} We complete the proof using the terms involved in the fourth-order case ($m=4$) because it simplifies the indexing tremendously although the terms in our proof will be entirely general. Consider the error at the $(k+1)$\\textsuperscript{th} iteration: \n\\[\\begin{aligned}\n \\vx\\itn{k+1} - \\vx & = \\frac{\\alpha}{m-1} \\mR [ (\\mI \\kron \\vx\\itn{k} \\kron \\vx\\itn{k}  + \\vx\\itn{k} \\kron \\mI \\kron \\vx\\itn{k} + \\vx\\itn{k} \\kron \\vx\\itn{k} \\kron \\mI ) \\vx\\itn{k+1} \\\\\n & \\qquad \\qquad \\qquad - (\\mI \\kron \\vx \\kron \\vx + \\vx \\kron \\mI \\kron \\vx + \\vx \\kron \\vx \\kron \\mI ) \\vx ] \\\\ \n  & = \\frac{\\alpha}{m-1} \\mR [ (\\vx\\itn{k+1} \\kron \\vx\\itn{k} \\kron \\vx\\itn{k}  + \\vx\\itn{k} \\kron \\vx\\itn{k+1} \\kron \\vx\\itn{k} + \\vx\\itn{k} \\kron \\vx\\itn{k} \\kron \\vx\\itn{k+1} ) \\\\\n  & \\qquad \\qquad \\qquad - ( \\vx \\kron \\vx \\kron \\vx + \\vx \\kron \\vx \\kron \\vx + \\vx \\kron \\vx \\kron \\vx). ]\n  \\end{aligned} \\]\nAt this point, it suffices to prove that terms of the form $\\normof[1]{ \\vx\\itn{k+1} \\kron \\vx\\itn{k} \\kron \\vx\\itn{k} - \\vx \\kron \\vx \\kron \\vx } $ are bounded by $ ( m-1) \\normof[1]{ \\vx\\itn{k+1} - \\vx} + (m-1)(m-2) \\normof[1]{\\vx\\itn{k} - \\vx}$. Showing this for one term also suffices because all of these terms are equivalent up to a permutation.\n\nWe continue by Lemma~\\ref{lem:stochastic-diff}, which yields \n\\[ \\normof[1]{ \\vx\\itn{k+1} \\kron \\vx\\itn{k} \\kron \\vx\\itn{k} - \\vx \\kron \\vx \\kron \\vx } \\le \\normof[1]{ \\vx\\itn{k+1} - \\vx} + 2 \\normof[1]{ \\vx\\itn{k} - \\vx}  \\]\nin the third-order case, and $\\normof[1]{ \\vx\\itn{k+1} - \\vx} + (m-2) \\normof[1]{ \\vx\\itn{k} - \\vx}$ in general. Since there are $m-1$ of these terms, we are done.\n\\end{proof}\n\nIn comparison to the inner-outer iteration, this method requires detailed knowledge of the operator $\\cmP$ in order to form $\\mS(\\vx_k)$ or even matrix-vector products $\\mS(\\vx_k) \\vz$. In some applications this may be easy. In Figure~\\ref{fig:inverse}, we illustrate the convergence of the inverse iteration on the problems that the inner-outer method's illustration used. The convergence pattern is the same. \n\n\\begin{figure}\n\\includegraphics[width=0.5\\linewidth]{inverse-R4_11-97}%\n\\includegraphics[width=0.5\\linewidth]{inverse-R4_11-99}%\n\n\\caption{At left, the iterates from the inverse iteration to solve problem $\\mR_2$ with $\\alpha = 0.97$ and at right, the iterates to solve problem $\\mR_2$ with $\\alpha = 0.99$. Both show similar convergence behavior as to the inner-outer method.}\n\\label{fig:inverse}\n\\end{figure}\n\n\n\\subsection{Newton's method}\nFinally, consider Newton's method for solving the nonlinear equation: \n\\[ \\vf(\\vx) = \\alpha \\mR (\\vx \\kron \\vx) + (1-\\alpha) \\vv - \\vx = 0. \\]\nThe Jacobian of this operator is: \n\\[ \\mJ(\\vx) = \\alpha \\mR (\\vx \\kron \\mI + \\mI \\kron \\vx) - \\mI. \\]\n\nWe now prove the following theorem about the convergence of Newton's method.\n\n\\begin{theorem} \\label{thm:newton}\nLet $\\cmP$ be a third-order stochastic tensor, let $\\vv$ be a stochastic vector, and let $\\alpha < 1\/2$. Let $\\mR$ be the flattening of $\\cmP$ along the first index. Let \n\\[ \\vf(\\vx) = \\alpha \\mR (\\vx \\kron \\vx) + (1-\\alpha) \\vv - \\vx = 0. \\] \nNewton's method to solve $\\vf(\\vx) = 0$, and hence compute the unique multilinear PageRank vector, is the iteration:\n\\begin{equation} \\label{eq:newton}\n \\left[ \\mI - \\alpha \\mR (\\vx\\itn{k} \\kron \\mI + \\mI \\kron \\vx\\itn{k}) \\right] \\vp\\itn{k} = \\vf(\\vx\\itn{k}) \\qquad \\vx\\itn{k+1} = \\vx\\itn{k} + \\vp\\itn{k} \\qquad \\vx\\itn{0} = 0.\n\\end{equation}\nIt produces a unique sequence of iterates where: \n\\[ \\vf(\\vx\\itn{k}) \\ge 0 \\quad \\ve^{T} \\vf(\\vx\\itn{k}) = \\frac{\\alpha (\\ve^{T} \\vf(\\vx\\itn{k-1}))^2}{(1-2\\alpha)^2 + 4 \\alpha \\ve^{T} \\vf(\\vx\\itn{k-1})} \\le \\alpha (1-\\alpha)^2 \\frac{1}{4^{k-1}} \\quad k \\ge 1  \\]\nthat also converges quadratically in the $k\\to\\infty$ limit.\n\\end{theorem}\n\nThis result shows that Newton's method always converges quadratically fast when solving multilinear PageRank vectors inside the unique regime.\n\n\\begin{proof}\n We outline the following sequence of facts and lemmas we provide to compute the result. The key idea is to use the result that second-order multilinear PageRank is a 2nd-degree polynomial, and hence, we can use Taylor's theorem to derive an \\emph{exact} prediction of the function value at successive iterations. We first prove this \\emph{key fact}. Subsequent steps of the proof establish that the sequence of iterates is unique and well-defined (that is, that the Jacobian is always non-singular). This involves showing, additionally, that $\\vx\\itn{k} \\ge 0$, $\\ve^{T} \\vx\\itn{k} \\le 1$, and $\\vf(\\vx\\itn{k}) \\ge 0$. Let $f_k = \\ve^{T} \\vf$, since $\\vf(\\vx\\itn{k}) \\ge 0$, showing that $f_k \\to 0$ suffices to show convergence. Finally, we derive a recurrence: \\begin{equation} \\label{eq:newton-recur} f_{k+1} = \\frac{\\alpha f_k^2}{(1-2\\alpha)^2 + 4 \\alpha f_k}. \\end{equation}\n \n\\noindent \\textbf{Key fact.}  Let $\\vp\\itn{k} = \\vx\\itn{k+1} - \\vx\\itn{k}$. If the Jacobian $\\mJ(\\vx\\itn{k})$ is non-singular in the $k$\\textsuperscript{th} iteration, \nthen $\\vf(\\vx\\itn{k+1}) = \\alpha \\mR (\\vp\\itn{k} \\kron \\vp\\itn{k})$. To prove this, we use an exact version of Taylor's theorem around the point $\\vx\\itn{k}$: \n\\[ \\vf(\\vx\\itn{k} + \\vp) = \\vf(\\vx\\itn{k}) + \\mJ(\\vx\\itn{k}) \\vp + \\tfrac{1}{2} \\cmT \\vp^2, \\]\nwhere $\\cmT \\vp^2 = \\alpha \\mR (\\mI \\kron \\mI + \\mI \\kron \\mI) (\\vp \\kron \\vp)$ is \\emph{independent} of the current point. Note also that Newton's method chooses $\\vp$ such that $\\vf(\\vx\\itn{k}) + \\mJ(\\vx\\itn{k}) \\vp = 0$.  Then\n\\[ \\vf(\\vx\\itn{k+1}) =  \\vf(\\vx\\itn{k} + \\vp\\itn{k}) = \\alpha \\mR (\\vp\\itn{k} \\kron \\vp\\itn{k}). \\]\n\n\\noindent \\textbf{Well-defined sequence.} We now show that $\\mJ(\\vx\\itn{k})$ is non-singular for all $\\vx\\itn{k}$, and hence, the Newton iteration is well-defined. It is easy to do so if we establish that\n\\begin{equation} \\label{eq:newton-props}\n \\vf(\\vx\\itn{k}) \\ge 0, \\vx\\itn{k} \\ge 0, \\text{ and } z_k = \\ve^{T} \\vx\\itn{k} \\le 1 \n\\end{equation}\nalso holds at each iteration. Clearly, these properties hold for the initial iteration where $\\vf(\\vx\\itn{0}) = (1-\\alpha) \\vv$. Thus, we proceed inductively.  Note that if $\\vx\\itn{k} \\ge 0$ and $z_k \\le 1$ then the Jacobian is non-singular because $-\\mJ(\\vx\\itn{k}) = [\\mI - \\alpha \\mR ( \\vx\\itn{k} \\kron \\mI + \\mI \\kron \\vx\\itn{k} ) ]$ is a strictly diagonally dominant matrix, $M$-matrix when $\\alpha < 1\/2$. (In fact, both $\\mR ( \\vx\\itn{k} \\kron \\mI)$ and $\\mR (\\mI \\kron \\vx\\itn{k})$ are nonnegative matrices with column norms equal to $z_k = \\ve^{T}\\vx\\itn{k}$.) Thus, $\\vx\\itn{k+1}$ is well-defined and it remains to show that $\\vf(\\vx\\itn{k+1}) \\ge 0$, $\\vx\\itn{k+1} \\ge 0$, and $z_{k+1} \\le 1$. Now, by the definition of Newton's method: \n\\[ \\vx\\itn{k+1} = \\vx\\itn{k} - \\mJ(\\vx\\itn{k})^{-1} \\vf(\\vx\\itn{k}), \\] but $-\\mJ$ is an $M$-matrix, and so $\\vx\\itn{k+1} \\ge 0$. This also shows that $\\vp\\itn{k} = \\vx\\itn{k+1} - \\vx\\itn{k} \\ge 0$, from which, we can use our key fact to derive that $\\vf(\\vx\\itn{k+1}) \\ge 0$. What remains to show is that $z_{k+1} \\le 1$. By taking summations on both sides of \\eqref{eq:newton}, we have: \n\\[ (1-2\\alpha z_k) (z_{k+1} - z_k) = \\alpha z_k^2 + (1-\\alpha) - z_k. \\]\nA quick, but omitted, calculation confirms that $z_{k+1} > 1$ implies $z_{k} > 1$. Thus, we completed our inductive conditions for \\eqref{eq:newton-props}.\n\n\\noindent \\textbf{Recurrence} We now show that \\eqref{eq:newton-recur} holds. First, observe that \n\\[ f_k = \\alpha (\\ve^{T} \\vp\\itn{k})^2 \\qquad \\ve^{T} \\vp\\itn{k} = \\frac{f_k}{1-2 \\alpha z_k} \\qquad \\alpha z_k^2 + (1-\\alpha) - z_k - f_k = 0. \\]\nWe now solve for $z_k$ in terms of $f_k$. This involves picking a root for the quadratic equation. Since $z_k \\le 1$, this makes the choice the negative root in: \n\\[ z_k = \\frac{1 - \\sqrt{ (1-2\\alpha)^2 + 4 \\alpha f_k }}{2 \\alpha} \\le 1. \\] Assembling these pieces yields \\eqref{eq:newton-recur}.\n\n\\noindent \\textbf{Convergence} We have an easy result that $f_{k+1} \\le \\frac{1}{4} f_k$ by ignoring the term $(1-2\\alpha)^2$ in the denominator. Also, by direct evaluation, $f_1 = \\alpha (1-\\alpha)^2$. Thus, \n\\[ f_{k} \\le \\frac{1}{4^{k-1}} f_1 = \\frac{1}{4^{k-1}} \\alpha (1-\\alpha)^2, \\] which is one side of the convergence rate. The sequence for $f_k$ also converges quadratically in the limit because $\\lim_{k \\to \\infty} \\frac{f_{k+1}}{f_{k}^2}  = \\frac{\\alpha}{(1-2 \\alpha)^2}$.\n\\end{proof}\n\n\\paragraph{A practical, always-stochastic Newton iteration}\nThe Newton iteration from Theorem~\\ref{thm:newton} begins as $\\vx\\itn{0} = 0$ and, when $\\alpha < 1\/2$, gradually grows the solution until it becomes stochastic and attains optimality. For problems when $\\alpha > 1\/2$, however, this iteration often converges to a fixed point where $\\vx$ is not stochastic. (In fact, it always did this in our brief investigations.) To make our codes practical for problems where $\\alpha > 1\/2$, then, we enforce an explicit stochastic normalization after each Newton step:\n\\begin{equation} \\label{eq:newton-stochastic}\n[\\eye - \\alpha \\mR(\\vx\\itn{k} \\kron \\mI + \\mI \\kron \\vx\\itn{k})] \\vp\\itn{k} = \\vf(\\vx\\itn{k}) \\quad \\vx\\itn{k+1} = \\text{proj}(\\vx\\itn{k} + \\vp\\itn{k}) \\quad \\vx\\itn{0} = (1-\\alpha) \\vv,\n\\end{equation}\nwhere \n\\[ \\text{proj}(\\vx) = \\max(\\vx,0) \/ \\ve^{T} \\max(\\vx,0) \\]\nis a projection operator onto the probability simplex that sets negative elements to zero and then normalizes those left to sum to one.  We found this iteration superior to a few other choices including using a proximal point projection operator to produce always stochastic iterates~\\cite[\\S6.2.5]{Parikh-2014-prox}. We illustrate an example of the difference in Figure~\\ref{fig:newton-stochastic} where the always-stochastic iteration solve the problem and the iteration without this projection converges to a non-stochastic fixed-point. Note that, like a general instance of Newton's method, the system $[\\eye - \\alpha \\mR(\\vx\\itn{k} \\kron \\mI + \\mI \\kron \\vx\\itn{k})]$ may be singular. We never ran into such a case in our experiments and in our study.\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\linewidth]{vanilla-newton}%\n\\includegraphics[width=0.5\\linewidth]{stochastic-newton}%\n\n\\caption{For problem $\\mR_2$ with $\\alpha = 0.99$, the Newton iteration from Theorem~\\ref{thm:newton} (left figure) converges to a non-stochastic solution --- note the scale of the solution axis. The always-stochastic iteration~\\eqref{eq:newton-stochastic} (right figure) converges for this problem. Thus, we recommend the iteration~\\eqref{eq:newton-stochastic} when $\\alpha > \\frac{1}{m-1}$.}\n\\label{fig:newton-stochastic}\n\\end{figure}\n\n\nWe further illustrate the behavior of Newton's method on $\\mR_2$ with $\\alpha = 0.97$ and $\\mR_2$ with $\\alpha = 0.99$ in Figure~\\ref{fig:newton}. The second of these problems did not converge for either the inner-outer or inverse iteration. Newton's method solves it in just a few iterations. In comparison to both the inner-outer and inverse iteration, however, Newton's method requires even more direct access to $\\cmP$ in order to solve for the steps with the Jacobian.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\linewidth]{newton-R4_11-97}%\n\\includegraphics[width=0.5\\linewidth]{newton-R4_11-99}%\n\n\\caption{At left, the iterates of Newton's method to solve problem $\\mR_2$ with $\\alpha = 0.97$ and at right, the iterates to solve problem $\\mR_2$ with $\\alpha = 0.99$.  Both sequences converge unlike the inner-outer and inverse iterations.}\n\\label{fig:newton}\n\\end{figure}\n\n\\section{Experimental Results}\n\\label{sec:experiments}\n\nTo evaluate these algorithms, we create a database of problematic tensors. We then use this database to address two questions. \n\\begin{enumerate}\n\\item What value of the shift is most reliable?\n\\item Which method has the most reliable convergence?\n\\end{enumerate}\nIn terms of reliability, we wish for the method to generate a residual smaller than $10^{-8}$ before reaching the maximum iteration count where the residual for a potential solution is given by~\\eqref{eq:residual}. In many of the experiments, we run the methods for between 10,000 to 100,000 iterations. If we do not see convergence in this period, we deem a particular trial a failure. The value of $\\vv$ is always $\\ve \/n$ but $\\alpha$ will vary between our trials.\nBefore describing these results, we begin by discussing how we created the test problems.\n\n\\subsection{Problems}\nWe used exhaustive enumeration to identify $2 \\times 2 \\times 2$ and $3 \\times 3 \\times 3$ binary tensors, which we then normalized to stochastic tensors, that exhibited convergence problems with the fixed point or shifted methods. We also randomly sampled many $4 \\times 4 \\times 4$ binary problems and saved those that showed slow or erratic convergence for these same algorithms. We used $\\alpha = 0.99$ and $\\vv = \\ve\/n$ for these studies. The $6 \\times 6 \\times 6$ problems were constructed randomly in an attempt to be adversarial. Tensors with strong ``directionality'' seemed to arise as interesting cases in much of our theoretical study (this is not presented here). By this, we mean, for instance, tensors where a single state has many incoming links. We created a random procedure that generates problems where this is true (the exact method is in the online supplement) and used this to generate $6 \\times 6 \\times 6$ problems. In total, we have the following problems: \n\\[ \\begin{aligned}\n3 \\times 3 \\times 3 & \\qquad \\text{5 problems} \\\\\n4 \\times 4 \\times 4 & \\qquad \\text{19 problems} \\\\\n6 \\times 6 \\times 6 & \\qquad \\text{5 problems}.\n\\end{aligned} \\]\nThe full list of problems is in given in Appendix~\\ref{app:problems}.\n\nWe used Matlab's symbolic toolbox to compute a set of exact solutions to these problems. These $6 \\times 6 \\times 6$ problems often had multiple solutions whereas the smaller problems only had a single solution (for the values of $\\alpha$ we considered). While it is possible there are solutions missed by this tool, prior research found symbolic computation a reliable means of solving these polynomial systems of equations~\\cite{Kolda-2011-sshopm}.\n\n\\subsection{Shifted iteration}\n\\label{sec:shiftstudy}\nWe begin our study by looking at a problem where the necessary shift suggested by the ODE theory ($\\gamma = 1\/2$ for third-order data) does not result in convergence. We are interested in whether or not varying the shift will alter the convergence behavior. This is indeed the case. For the problem $\\mR_{4,11}$ from the appendix with $\\alpha = 0.99$, we show the convergence of the residual as the shift $\\gamma$ varies in Figure~\\ref{fig:shiftstudy}. When $\\gamma = 0.5$, the iteration does not converge. There is a point somewhere between $\\gamma = 0.554$ and $\\gamma = 0.5545$ where the iteration begins to converge. When we set $\\gamma = 1$, the iteration converged rapidly. \n\nIn the next experiment, we wished to understand how the reliability of the method depended on the shift $\\gamma$. In Table~\\ref{tab:shifttable}, we vary $\\alpha$ and the shift $\\gamma$ and look at how many of the $29$ test problems the shifted method can solve within 10,000 iterations. Recall that a method solves a problem if it pushes the residual below $10^{-8}$ within the iteration bound. The results from that table show that $\\gamma = 1$ or $\\gamma = 2$ results in the most reliable method. When $\\gamma = 10$, then the method was less reliable. This is likely due to the shift delaying convergence for too long. Note that we chose many of the problems based on the failure of the shifted method with $\\gamma = 0$ or $\\gamma = 1\/2$ and so the poor performance of these choices may not reflect their true reliability. Nevertheless, based on the results of this table, we recommend a shift of $\\gamma = 1$ for a third-order problem, or a shift of $\\gamma = m-2$ for a problem with an order-$m$ tensor. \n\n\\begin{figure}\n\\centering\n \\includegraphics{shiftstudy_R4_19}\n \\caption{For the problem $\\mR_{4,11}$ with $\\alpha = 0.99$ and $\\vv = \\ve\/n$, the shifted method will not converge unless $\\gamma$ is slightly larger than 0.554. As $\\gamma$ becomes larger, the convergence rate increases.}\n \\label{fig:shiftstudy}\n\\end{figure}\n\n\\begin{table}\n\\caption{Each row of this table reports the number of problems successfully solved by the shifted iteration as the shift varies from $0$ to $10$. The values are reported for each value of $\\alpha$ considered, as well as broken down into the different problem sizes considered.   }\n\\label{tab:shifttable}\n \\centering\n \\begin{tabularx}{0.7\\linewidth}{>{\\hsize=1.35\\hsize}X >{\\hsize=1.35\\hsize}X *{7}{>{\\hsize=0.9\\hsize}X}}\n  \\toprule\n$\\alpha$ & $n$ & \\multicolumn{7}{l}{Shifts $\\gamma$}  \\\\\n&     & 0 & 1\/4 & 1\/2 & 3\/4 & 1 & 2 & 10 \\\\\n\\midrule\n  0.70  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 19  & 19  & 19  & 19  & 19  & 19  & 19  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 29  & 29  & 29  & 29  & 29  & 29  & 29  \\\\ \\midrule\n 0.85  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 19  & 19  & 19  & 19  & 19  & 19  & 19  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 29  & 29  & 29  & 29  & 29  & 29  & 29  \\\\ \\midrule\n 0.90  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 18  & 19  & 19  & 19  & 19  & 19  & 19  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 28  & 29  & 29  & 29  & 29  & 29  & 29  \\\\ \\midrule\n 0.95  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 7  & 11  & 13  & 13  & 16  & 19  & 18  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 17  & 21  & 23  & 23  & 26  & 29  & 28  \\\\ \\midrule\n 0.99  &  3  & 4  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 0  & 1  & 1  & 2  & 2  & 2  & 2  \\\\ \n       &  6  & 1  & 1  & 1  & 2  & 2  & 2  & 1  \\\\ \\addlinespace \n       &     & 5  & 7  & 7  & 9  & 9  & 9  & 8  \\\\ \n\\bottomrule\n\\end{tabularx}\n\\end{table}\n\n\n\\subsection{Solver reliability}\n\\label{sec:solvers-perf}\n\nIn our final study, we utilize each method with the following default parameters: \n\\begin{center}\n \\begin{tabular}{lll}\n  F & fixed point & 10,000 maximum iterations, $\\vx_0 = \\vv$  \\\\\n  S & shifted & 10,000 maximum iterations, $\\gamma = 1$, $\\vx_0 = \\vv$  \\\\\n  IO & inner-outer & 1,000 outer iterations, internal tolerance $\\eps$, $\\vx_0 = \\vv$  \\\\\n  Inv & inverse & 1,000 iterations, $\\vx_0 = \\vv$  \\\\\n  N & Newton & 1,000 iterations, projection step, $\\vx_0 = (1-\\alpha) \\vv$ \n \\end{tabular}\n\\end{center}\nWe also evaluate each method with $10$ times the default number of iterations. \n\nThe results of the evaluation are shown in Table~\\ref{tab:methods} as $\\alpha$ varies from $0.7$ to $0.99$. The fixed point method has the worst performance when $\\alpha$ is large. Curiously, when $\\alpha = 0.99$ the shifted method outperforms the inverse iteration, but when $\\alpha = 0.95$ the inverse iteration outperforms the shifted iteration. This implies that the behavior and reliability of the methods is not monotonic in $\\alpha$. While this fact is not overly surprising, it is pleasing to see a concrete example that might suggest some tweaks to the methods to improve their reliability. Overall, the inner-outer and Newton's method have the most reliable convergence on these difficult problems.\n\n\n\\begin{table}\n\\caption{Each row of this table reports the number of problems successfully solved by the various iterative methods in two cases: with their default parameters, and with 10 times the standard number of iterations. The values are reported for each value of $\\alpha$ considered, as well as broken down into the different problem sizes considered. The columns are: F for the fixed-point, S for the shifted method, IO for the inner-outer, Inv for the inverse iteration, and N for Newton's method. }\n\\label{tab:methods}\n\\centering\n\\begin{tabularx}{\\linewidth}{>{\\hsize=1.5\\hsize}X >{\\hsize=1.5\\hsize}X *{5}{>{\\hsize=0.9\\hsize}X} @{\\qquad\\qquad} *{5}{>{\\hsize=0.9\\hsize}X} }\n\\toprule\n$\\alpha$ & $n$ & \\multicolumn{5}{l}{Method (defaults)} & \\multicolumn{5}{l}{Method (Extra iteration)} \\\\\n         &     & F & S & IO & Inv & N  & F & S & IO & Inv & N\\\\\n\\midrule\n 0.70  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 19  & 19  & 19  & 19  & 19  & 19  & 19  & 19  & 19  & 19  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 29  & 29  & 29  & 29  & 29  & 29  & 29  & 29  & 29  & 29  \\\\ \\midrule \n 0.85  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 19  & 19  & 19  & 19  & 19  & 19  & 19  & 19  & 19  & 19  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 29  & 29  & 29  & 29  & 29  & 29  & 29  & 29  & 29  & 29  \\\\ \\midrule \n 0.90  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 18  & 19  & 19  & 19  & 19  & 18  & 19  & 19  & 19  & 19  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 28  & 29  & 29  & 29  & 29  & 28  & 29  & 29  & 29  & 29  \\\\ \\midrule \n 0.95  &  3  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 7  & 16  & 18  & 19  & 19  & 8  & 16  & 19  & 19  & 19  \\\\ \n       &  6  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  & 5  \\\\ \\addlinespace \n       &     & 17  & 26  & 28  & 29  & 29  & 18  & 26  & 29  & 29  & 29  \\\\ \\midrule \n 0.99  &  3  & 4  & 5  & 5  & 5  & 5  & 4  & 5  & 5  & 5  & 5  \\\\ \n       &  4  & 0  & 2  & 15  & 1  & 19  & 0  & 2  & 17  & 1  & 19  \\\\ \n       &  6  & 1  & 2  & 3  & 1  & 4  & 2  & 3  & 4  & 3  & 4  \\\\ \\addlinespace \n       &     & 5  & 9  & 23  & 7  & 28  & 6  & 10  & 26  & 9  & 28  \\\\  \n\\bottomrule\t\t\t\n\\end{tabularx}\n\\end{table}\n\nNewton's method, in fact, solves all but one instance: $\\mR_{6,3}$ with $\\alpha = 0.99$. We explore this problem in slightly more depth in Figure~\\ref{fig:newton-failure}. This problem only has a single unique solution (based on our symbolic computation). However, none of the iterations will find it using the default settings --- all the methods are attracted to a point with a small residual and an indefinite Jacobian. We were able to find the true solution by using Newton's method with random starting points. It seems that the iterates need to approach the solution from on a rather precise trajectory in order to overcome an indefinite region. This problem should be a useful case for future algorithmic studies on the problem. \n\n\\begin{figure}\n\\centering\n \\includegraphics{newton_failure_R6_3}%\n\\begin{minipage}[b]{0.6\\linewidth}\n\n\\normalsize The point ~~~~~~~~~ The eigenvalues ~~ The true solution\n\n\\scriptsize%\n\\begin{verbatim}\n 0.199907259533067   0.980000000000000   0.043820721946272\n 0.006619352098700   0.000064771773360   0.002224192630620\n 0.116429656827957  -1.786544142144891   0.009256490884022\n 0.223220491129316  -0.575965838505486   0.819168263512464\n 0.079958855790239  -0.575965838505486   0.031217440669761\n 0.373864384620721  -1.438690261635567   0.094312890356862\n\\end{verbatim}\n\\bigskip\n\n\\normalsize The Jacobian\n\n\\scriptsize\n\\begin{verbatim}\n -0.9712    0.2246    0.3496    0.1944    0.3395    0.7435\n       0   -0.7299    0.0131         0    0.0824         0\n  0.4781    0.1851   -0.9505         0    0.4621    0.2408\n  0.0288    0.1851    0.0495    0.1822         0    0.4453\n       0    0.4192    0.3701    0.0857   -0.5939    0.1581\n  1.4443    0.6960    1.1482    0.5176    0.6899   -0.6077\n\\end{verbatim}\\bigskip\n\n\n\n\\end{minipage}%\n\n\\caption{Newton's method on the non-convergent case of $\\mR_{6,3}$ with $\\alpha = 0.99$. The method repeatedly drops to a small residual before moving away from the solution. This happens because when the residual becomes small, then the Jacobian acquires a non-trivial positive eigenvalue as exemplified by the point with the red-circled residual. We show the Jacobian and eigenvalues at this point. It appears to be a pseudo-solution that attracts all of the algorithms. Using random starting points, Newton's method will sometimes generate the true solution, which is far from the the attracting point.}\n\\label{fig:newton-failure}\n\\end{figure}\n\n\n\\section{Discussion}\n\nIn this manuscript, we studied the higher-order PageRank problem as well as the multilinear PageRank problem. The higher-order PageRank problem behaves much like the standard PageRank problem: we always have guaranteed uniqueness and fast convergence. The multilinear PageRank problem, in contrast, only has uniqueness and fast convergence in a more narrow regime. Outside of that regime, existence of a solution is guaranteed, although uniqueness is not. As we were finalizing our manuscript for submission, we discovered an independent preprint that discusses some related results from an eigenvalue perspective~\\cite{Chu-2014-eigenvalue}. \n\nFor the multilinear PageRank problem, convergence of an iterative method outside of the uniqueness regime is highly dependent on the data. We created a test set based on problems where both the fixed-point and shifted fixed-point method fails. On these tough problems, both the inner-outer and Newton iterations had the best performance. This result suggests a two-phase approach to solving the problems: first try the simple shifted method. If that does not seem to converge, then use either a Newton or inner-outer iteration. Our empirical findings are limited to the third order case and we plan to revisit such strategies in the future when we consider large scale implementations of these methods on real-world problems --- the present efforts are focused on understanding what is and is not possible with the multilinear PageRank problem. This is also due to the observation that the multilinear PageRank problem is only interesting for massive problems. If $O(n^2)$ memory is available, then the higher-order PageRank vector should be used instead, unless there is a modeling reason to choose the multilinear PageRank formulation. \n\nBased on our theoretical results, we note that there seems to be a key transition for \\emph{all} of the algorithms and theory that arises at the uniqueness threshold: $\\alpha < 1\/(m-1)$. We are currently trying to find algorithms with guaranteed convergence when $\\alpha > 1\/(m-1)$ but have not been successful yet. We plan to explore using sum-of-squares programming for this task in the future. Such an approach has given one of the first algorithms with good guarantees for the tensor eigenvalue problem~\\cite{Nie-2013-sdp}.\n \n\\section*{Acknowledgments}\nWe are grateful to Austin Benson for suggesting an idea that led to the stochastic process as well as some preliminary comments on the manuscript. DFG would like to acknowledge support for NSF CCF-1149756. LHL gratefully acknowledges support for AFOSR FA9550-13-1-0133, NSF DMS-1209136, and NSF DMS-1057064.\n\n\\bibliographystyle{dgleich-bib}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLight-element abundance inhomogeneities in globular clusters have been\na subject of active study for the past thirty years. The phenomenon\nwas first noticed as anticorrelated variations in the broad CN and CH\nabsorption features in red giants in a few clusters, and with the\ndevelopment of larger-aperture telescopes and higher-resolution\\linebreak\nspectrographs, the data set has expanded in several\ndirections. Star-to-star abundance variations have been found in stars\nfrom the main sequence to the tip of the red giant branch, the\nset of abundances involved has expanded to include carbon, nitrogen,\noxygen, sodium, magnesium and aluminium, and individual studies often survey\ntens of clusters rather than one or two. \n\nIntriguingly, stars with these unusual abundances are\\linebreak found\nuniversally in globular clusters, but are apparently only formed in\nthat environment. The overall picture that has developed is that stars\nat all evolutionary phases in all Galactic globular clusters occupy a wide\nrange in light-\\linebreak element abundance that is not observed in any other\nGalactic stellar population, or in the fields of Local Group dwarf\ngalaxies. This abundance range is typically observed in anticorrelated\nabundance pairs, C vs N or O vs Na, because roughly half of cluster\nstars have abundance patterns like Population II field stars while the\nother half are relatively depleted in carbon, oxygen and magnesium and\nenhanced in nitrogen, sodium and aluminium, with no variations in any\nother elemental abundances.\n\nThere are a few globular clusters known to exhibit\\linebreak metallicity\nvariations along with light-element variations, such as $\\omega$ Cen\nand M54, and they are unusual in other ways as well, with large masses\nand likely extragalactic origins (e.g., Carretta et\nal. 2010\\nocite{CBG10}). However, they are an exception, while\nclusters with only light-element variations appear to be the\nrule. Comparative studies of light-element abundances in globular\ncluster stars and halo field stars (e.g., Langer, Suntzeff \\& Kraft\n1992\\nocite{LSK92}) find that the field star population shows very\nlittle or no light-element abundance variation. The field-star studies\nof Pilachowski et al. (1996a)\\nocite{PSK96} and Gratton et al. (2000)\\nocite{GSCB00} found no stars with cluster-like\nabundance variations in the field, and concluded that the cluster\nenvironment must play a vital role in creating or permitting light-element\nabundance variations. \n\nA similar paucity of stars with cluster-like light-element abundances has been\nreported in nearby dwarf galaxies.\\linebreak McWilliam \\& Smecker-Hane (2005)\\nocite{MWSH05} and Sbordone et al. (2007)\\nocite{SBB07}\nboth report quite unusual abundance patterns in the Sagittarius dwarf\ngalaxy, but no stars with cluster-like C-N or O-Na\nanticorrelations. Shetrone et al. (2003)\\nocite{SVT03} surveyed stars in Sculptor, Fornax,\nCarina and Leo I and found none with cluster-like light-element\nabudnaces, and Letarte et al. (2010)\\nocite{LHT10} confirm that result with more stars in\nFornax. More recently, Maretll \\& Grebel (2010)\\nocite{MG10} searched a sample of\nfield giants from the SEGUE survey (Yanny et al. 2009)\\nocite{Y09} and\\linebreak found that $2.5\\%$\nof those stars have strong CN and weak CH features relative to the\nmajority of the field at the same metallicity and luminosity,\nsuggesting that they may have the full cluster-like light-element\nabundance pattern. Those authors claim that the CN-strong field stars\nformed in globular clusters and later migrated into the halo as a\nresult of cluster mass-loss and dissolution processes. \n\nStudies of Galactic globular cluster abundances are experiencing\nsomething of a revival at present, driven by the discovery that the\n(presently) most massive clusters have complex color-magnitude\ndiagrams (CMDs). There is a variety of unexpected behavior found in\ncarefully constructed, highly accurate CMDs: multiple main sequences\nin M54 (e.g., Siegel et al. 2007\\nocite{SDM07}; Carretta et\nal. 2010\\nocite{CBG10}), $\\omega$ Centauri (e.g., Bedin et\nal. 2004\\nocite{BPA04}; Sollima et al. 2007\\nocite{SFB07}; Bellini et\nal. 2010\\nocite{BBP10}) and NGC 2808 (Piotto et\nal. 2007\\nocite{PBA07}), multiple subgiant branches in M54, $\\omega$\nCen, 47 Tucanae (e.g., Anderson et al. 2009\\nocite{APK09}), NGC 1851\n(e.g., Cassisi et al. 2008\\nocite{CSP08}; Milone et\nal. 2008\\nocite{MBP08}; Yong \\& Grundahl 2008\\nocite{YG08}), NGC 6388\n(Piotto 2009\\nocite{P09}) and M22 (Piotto 2009), and broadened red giant\nbranches (RGBs) in M4 (Marino et al. 2008\\nocite{MVP08}) and NGC 6752\n(Milone et al. 2010\\nocite{MPK10}). \n\nThere are several tantalizing potential connections between the\nphotometric multiplicity in clusters and light-\\linebreak element abundance\nvariations. Following the suggestion in Cassisi et al. (2008)\\nocite{CSP08} that the two\ndistinct subgiant branches discovered in NGC 1851 by Milone et al. (2008)\\nocite{MBP08}\nhave very similar ages but total [C+N+O\/Fe] abundances that differ by\na factor of two, Yong et al. (2009)\\nocite{YGD09} measured abundances of carbon, nitrogen\nand oxygen in four bright red giants in NGC 1851. The total [C+N+O\/Fe]\nabundance in their data set does vary fairly widely, which is\nconsistent with a model in which NGC 1851 contains two populations in\ntotal light-element abundance. The case of NGC 1851 also provides a\nuseful illustration of the importance of a careful choice of\nphotometric systems: while the division of the subgiant branch is\nvisible in the Milone et al. (2008)\\nocite{MBP08} (F336W, F814W) and (F606W, F814W)\ncolor-magnitude diagrams, and in the Johnson-Cousins ($U,I$)\ncolor-\\linebreak magnitude diagram presented in Han et al. (2009b)\\nocite{HL09}, it is not visible\nin the Han et al. (2009b)\\nocite{HL09} ($V,I$) color-magnitude diagram or the \nStr\\\"{o}mgren ($vby$) color-magnitude diagram presented in Yong et al. (2009)\\nocite{YGD09}.\n\nIn M4, there is also RGB multiplicity that is found or not depending\non the filter set used to construct a color-magnitude diagram:\nMarino et al. (2008)\\nocite{MVP08} report a split of the red giant branch in the\nJohnson-Cousins ($U,B$) color-magnitude diagram, which had not been\nobserved in previous color-magnitude diagrams based on redder\nphotometric passbands. However, in M4, the two giant branches appear\nto correlate with typical globular cluster variations in [O\/Fe] and\n[Na\/Fe] abundances rather than variations in total [C+N+O\/Fe]\nabundance. \n\nIt is of course not surprising that abundance variations with\nsignificant spectral effects should have noticeable effects on\nphotometry, particularly in UV-blue photometric bands where CN and NH\nmolecular bands can be dominant. Photometric determinations of stellar\nparameters and iron abundance were the motivation for defining\nmedium-band filter systems like Str\\\"{o}mgren (Str\\\"{o}mgren 1963)\\nocite{S63}, DDO (McClure 1973)\\nocite{M73}\nand Washington (Canterna 1976)\\nocite{C76}. It is, however, unexpected that the\noxygen-sodium anticorrelation\\linebreak would affect photometry, particularly in\nthe very broad\\linebreak Johnson\/Cousins system, and it is unclear whether\nvariations in oxygen and sodium abundance, which are observed in all\nGalactic globular clusters (e.g., Carretta et al. 2009\\nocite{CBG09}),\ncorrelate with variations in $U-B$ color.\n\nThe complex phenomenology of photometrically multiple globular clusters,\nand the excitement about them, serve to underline the importance of\nthe rhetorical framework\\linebreak used in discussing an observed phenomenon: as\nan anomaly, star-to-star abundance variations are a curiosity to be\ncatalogued, but as a result of light-element self-enrichment and\nmultiple star-formation events they are powerful markers of the\nconditions in early Galactic history. The present challenge is to\nunderstand the connections between the new photometric observations\nand the well-studied abundance variations. This requires that we\ndevelop a comprehensive model for the origin of chemical and\nphotometric complexity in globular clusters, and further that we\nground that model in the larger cosmological environment to enable\nstudies of the effects of formation time and environment on the ability\nof a star cluster to self-enrich and to form a second stellar\ngeneration. \n\n\\section{Development of the observational data set}\nPhotographic color-magnitude diagrams constructed for\\linebreak globular\nclusters (e.g., Arp \\& Johnson 1955\\nocite{AJ55}; Sandage \\& Wallerstein\n1960\\nocite{SW60}) revealed simple stellar populations: unlike the\nwide variety found in surveys of the Solar neighborhood, stars in\nglobular and open clusters all apparently shared a single age and\nmetallicity. The clusters were\\linebreak quickly recognized as ideal\nlaboratories for testing theories of stellar structure and evolution\n(e.g., Sandage 1958\\nocite{S58}, Preston 1961\\nocite{P61}), and are\nused to the present day as anchors for metallicity scales (e.g., Kraft\n\\& Ivans 2003\\nocite{KI03}; Carretta et al. 2009\\nocite{CBG09b}).\n\n\\subsection{Carbon and nitrogen in bright red giants}\nIt was therefore suprising when these orderly, predictable stellar\nsystems turned out to host a number of stars with wide variations in\nthe strength of molecular features. Unusually weak absorption in the\nCH G band (the phenomenon of ``weak-G-band stars'') was noted among\ngiants in M92 (Zinn 1973\\nocite{Z73}; Butler, Carbon \\& Kraft\n1975\\nocite{BCK75}), in NGC 6397 (Mallia 1975\\nocite{M75}), in M13 and\nM15 (Norris \\& Zinn 1977\\nocite{NZ77}), in $\\omega$ Cen (Dickens \\&\nBell 1976\\nocite{DB76}), in M5 (Zinn 1977\\nocite{Z77}), and in 47 Tuc\n(Norris 1978\\nocite{N78}). It was quickly shown (Norris \\& Zinn 1977;\nZinn 1977) that most of the weak-G-band stars were on the asymptotic\ngiant branch, implying the existence of some process that dramatically\nreshapes the surface abundances of stars between the RGB and the AGB,\ndescribed as the ``third dredge-up'' by Iben (1975)\\nocite{I75}. \n\nIt was also noted (by, e.g., Norris \\& Cottrell 1979\\nocite{NC79};\nNorris et al. 1981\\nocite{N81}; Hesser et al. 1982\\nocite{HBH82};\nNorris, Freeman \\& Da Costa 1984\\nocite{N84}) that those RGB stars\nwith unusually weak CH bands also had relatively strong CN absorption\nat 3883\\hbox{\\AA} and 4215\\hbox{\\AA}. Since molecular abundance is\ncontrolled by the abundance of the minority species, CH traces carbon\nabundance, while CN reflects nitrogen abundance. This general\nassociation of bandstrength and abundance was confirmed by\nspectral-synthesis studies such as Bell \\& Dickens (1980)\\nocite{BD80}, which implies that\nthe CN-strong, CH-weak stars found\\linebreak only in globular clusters have\natmospheres that are depleted in carbon and enriched in nitrogen. \n\nSince the CNO cycle, operating in equilibrium, tends to convert both\ncarbon and oxygen into nitrogen, stars with strong CN and weak CH\nbands were interpreted as having some amount of CNO-processed material\nin their atmospheres. Several theories were put forward to explain\nthis extra CNO-cycle processing: McClure (1979)\\nocite{M79} surveyed the data\navailable at the time and concluded that internal mixing, specifically\nthe meridional circulation described by Sweigart \\& Mengel (1979)\\nocite{SM79}, could be\nresponsible for ``some or all'' of the surface abundance\nvariations. The connection between angular momentum and the efficacy\nof meridional circulation prompted Suntzeff (1981)\\nocite{S81} to propose that\ndifferent rotational velocities might explain the different levels of\ncarbon depletion seen in giants in M3 and M13, an idea further\nexplored in the Norris (1987)\\nocite{N87} study of the relation between CN anomalies\nand overall globular cluster ellipticity. Langer (1985)\\nocite{L85} suggested that a\nuniform mixing efficiency, in combination with star-to-star variations\nin CNO-cycle fusion rates, could produce the observed ranges in\nsurface carbon and nitrogen abundance. Cohen (1978)\\nocite{C78} proposed that\nstar-to-star scatter in [Na\/Fe] and [Ca\/Fe] in M3 giants required a\nnon-homogeneous initial gas cloud, a scenario that implies a\nprimordial origin for [C\/Fe] and [N\/Fe] variations as well. In\naddition, D'Antona et al. (1983)\\nocite{D83} proposed that light-element abundance variations\nwere merely surface pollution, a consequence of mass loss from evolved\nstars and the high density of globular clusters. \n\n\\subsection{Other elemental abundances}\nHoping to learn more about the source of apparently CNO-processed\nmaterial in the atmospheres of some globular\\linebreak cluster giants,\nresearchers began obtaining higher-\\linebreak resolution spectra for cluster\ngiants. These were difficult observations to make with the 4-m-class\ntelescopes available at the time, and as a result the data sets were\ntypically small and limited to stars brighter than $V\\simeq\n14$. Despite these challenges, it was quickly discovered that stars\ndepleted in carbon and enhanced in nitrogen were also depleted in\noxygen and magnesium, and enhanced in sodium and aluminium. CN-strong\ngiants in M5 were found by Sneden et al. (1992)\\nocite{S92} to have systematically lower\noxygen abundances and higher sodium abundances than their CN-normal\ncounterparts. Cottrell \\& Da Costa (1981)\\nocite{CD81} found positive correlations between CN\nbandstrength and both sodium and aluminium abundances in NGC 6752. A\ncorrelation\\linebreak between aluminium and sodium abundances, and\nanticorrelations between aluminium and both oxygen and magnesium, were\nfound by Shetrone (1996)\\nocite{S96} in M92, M13, M5 and M71, clusters that span the\nrange of halo globular cluster metallicity. \n\nOxygen depletion was consistent with an evolutionary explanation, as\nthe hydrogen-burning shell of a red giant is hot enough to host the\nCNO-cycle reactions $^{14}$N$+2$p$\\rightarrow$\\linebreak $^{12}$C$+\\alpha$. However,\nchanges in the abundances of heavier elements were unexpected from\nfusion reactions occurring within $0.8{\\rm M}_{\\odot}$ red giants: the NeNa\nand MgAl cycles operate similarly to the CNO cycle, with the\nnuclei acting as catalysts to convert hydrogen into helium, but both\nrequire significantly higher temperatures. Some authors (e.g.,\nPilachowski et al. 1996b\\nocite{P96}) interpreted the extension of the\nlight-element abundance variations to sodium, magnesium and aluminium\nas a sign that the hydrogen-burning shells in red giants must have the\nability to operate the hotter hydrogen-fusion cycles, while others\n(e.g., Peterson 1980\\nocite{P80}) considered it a sign that the\ninitial gas cloud from which globular clusters formed must have been\ninhomogeneous. \n\n\\subsection{Main-sequence and turnoff stars}\nWith the construction of 8-meter-class telescopes came access to\nfainter stars within globular clusters. Harbeck et al. (2003)\\nocite{HSG03} observed around\n100 stars at or below the main-sequence turnoff in 47 Tuc, and found\nclear bimodality in the distribution of CN bandstrength in those\nstars, implying variations in C and N abundance as large as those\nalready known in red giants in the cluster. Main-sequence and turnoff\nstars in M13 were observed by Briley et al. (2004)\\nocite{BCS04}, and significant,\nanticorrelated ranges in C and N abundance ($\\Delta$ [N\/Fe] $\\simeq\n1.0$, $\\Delta$ [C\/Fe] $\\simeq 0.5$) were also found among those stars.\n\nThese discoveries had a major impact on theoretical explanations for\nlight-element abundance variations in globular clusters, and prompted\na serious evaluation of the possibility that they are not simple\nstellar populations. While evolutionary explanations could conceivably be\nstretched to include modifications of surface aluminium abundance, they\ncould not accomodate abundance variations in low-mass main-sequence\nstars, which are not capable of either high-temperature hydrogen\nfusion or mixing between the surface and the core. Additionally, the\nfact that the abundance ranges are as large above the ``bump'' in the\nRGB luminosity function as below it indicates that the abundance\nvariations cannot be mere surface pollution, because such a signal\nwould be greatly diminished at the first dredge-up (Iben 1965)\\nocite{I65}, when \nthe surface convective zone briefly deepens well into the interior of \nthe star. \n\n\\section{Current models for globular cluster formation}\nThe presently favored explanation for the presence of primordial\nlight-element abundance variations in globular\\linebreak clusters is that the\nCN-strong, N- and Na-rich, C- and O-poor stars are a second generation\nformed from material processed by intermediate- or high-mass stars in\nthe first generation. There have been several types of stars proposed\nas the source of this feedback material, each with its own strengths\nand weaknesses. AGB stars with masses between $4$ and $8 {\\rm M}_{\\odot}$\n(e.g., Parmentier et al. 1999\\nocite{PJ99}) are popular because they\nare relatively common, they are known to have slow, massive winds,\nthey are a site of hot hydrogen burning, and they evolve on timescales\nof $\\sim 10^{8}$ years, quite fast compared to the lifetime of a\nglobular cluster. In addition to AGB stars, rapidly rotating massive\nstars (Decressin et al. 2007b)\\nocite{DM07} and massive binary stars undergoing mass transfer (De Mink et al. 2009)\\nocite{DM09} have been proposed as alternative sources of feedback\nmaterial, and both could deliver more feedback mass in a shorter\namount of time than AGB stars, though the stars themselves are less\ncommon. As is pointed out by Sills \\& Glebbeek (2010)\\nocite{SG10}, while it may be\nobservationally determined that one particular feedback source is\ndominant, they all certainly contribute to the cluster ISM at some\nlevel. \n\nIn a present-day globular cluster with a mass of $5\\times 10^{5}\n{\\rm M}_{\\odot}$ and a $1:1$ ratio of first- to second-generation stars,\nassuming a $30\\%$ star formation efficiency, the second generation of\nstars must have formed from $\\simeq 8\\times 10^{5} {\\rm M}_{\\odot}$ of\ngas. The first stellar generation, with a mass of $2.5\\times 10^{5}\n{\\rm M}_{\\odot}$, clearly cannot have produced enough feedback material to\nform the second generation (indeed, typical values for AGB mass loss\nare $\\le 10\\%$). There have been four solutions proposed for this\n``mass budget problem'': a top-heavy first-generation mass function\n(e.g., Decressin et al. 2007b\\nocite{DC07}), a second-generation mass\nfunction that is truncated above\\linebreak $0.8 {\\rm M}_{\\odot}$ (e.g., D'Ercole et\nal. 2008\\nocite{DVD08}), a first generation that is initially $10-20$ times as\nmassive as at present (e.g., D'Ercole et al. 2010\\nocite{DDV10}), and infall of\npristine gas (e.g., Carretta et al. 2010b\\nocite{CBG10b}; Conroy \\& Spergel 2010\\nocite{CS10}). \n\nA top-heavy first generation would alleviate the mass budget problem\nby placing more $5-15 {\\rm M}_{\\odot}$ feedback\\linebreak sources in the first\ngeneration.  A truncated (bottom-heavy) second generation would\nrequire the first generation to produce less feedback material:\nassuming a Kroupa IMF\\linebreak (Kroupa et al. 1993)\\nocite{KTG93} and a mass range from $0.1$ to\n$100 {\\rm M}_{\\odot}$, half of the mass is in stars with ${\\rm M} \\le\n0.8{\\rm M}_{\\odot}$. However, it is unclear what physical process would\ncause overproduction of massive stars in the first generation or\nunderproduction in the second, and neither effect is significant\nenough to solve the mass budget problem. Additionally, observations of\nyoung star clusters (e.g., Boudreault \\& Caballero 2010\\nocite{BC10};\nGennaro et al. 2010\\nocite{GB10}) do not find either of these effects,\nand present-day globular clusters do not contain unusually large\npopulations of neutron stars or other compact remnants of massive\nfirst-generation stars relative to the number of low-mass\nfirst-generation stars still on the main sequence (e.g., Bogdanov et\nal. 2010\\nocite{BvH10}; Lorimer 2010\\nocite{L10}).\n \n Massive first generations and significant gas infall, in contrast,\n are central to the leading theoretical models of globular cluster\n formation. In the model of D'Ercole et al. (2008)\\nocite{DVD08}, the first stellar\n generation has a mass $10$ to $20$ times its present mass, and\n produces all the material needed for the formation of the second\n generation from AGB winds. The authors assume that Type Ia supernovae\n begin occurring 40 Myr after the formation of the first\n generation. The supernovae conclusively end star formation in the\n cluster, meaning that all second-generation stars must have formed\n within that time. The second generation stars form near the cluster\n center, and as a result, when the cluster expands in response to the\n supernovae, the stars that become dissociated from the cluster are\n mostly or exclusively first-generation stars. The authors also\n consider a model with infall of pristine gas from near the globular\n cluster, and find that it prolongs star formation significantly\n beyond the onset of Ia supernovae. \n\nThe globular cluster formation model of Conroy \\& \\linebreak Spergel (2011)\\nocite{CS10} also relies on\nAGB stars to provide the\\linebreak chemical inhomogeneities between the first\nand second stellar generations, but requires significantly more gas\naccretion to provide the mass for second-generation stars. The authors\njustify this by speculating about the cosmological environment of\nproto-globular clusters, namely self-gravitating,\\linebreak gas-dominated\nproto-galactic systems with gas masses between $10^{8} {\\rm M}_{\\odot}$ and\n$10^{10} {\\rm M}_{\\odot}$. They find that globular clusters orbiting in such\nsystems can accrete significant amounts of gas over $10^{8}$ years,\nthrough a combination of Bondi accretion and ``sweeping up'' of\nmaterial in the cluster's path, with little vulnerability to\nram-pressure stripping for clusters above $10^{4}{\\rm M}_{\\odot}$. In this\nmodel, Lyman-Werner flux\\linebreak ($912 \\hbox{\\AA} \\le \\lambda \\le 1100\n\\hbox{\\AA}$) from massive first-generation stars prevents the\ngathering gas from forming stars by dissociating H$_{2}$ molecules,\ncreating a gap of roughly $10^{8}$ years between the two generations\nand allowing time for the mass in accreted gas to become large enough\nto create a second generation as massive as the first. \n\nAny globular cluster formation model that requires significant gas\naccretion implicitly assumes that the accreted gas must have a\nmetallicity very similar to that of the first stellar generation,\nsince systematic [Fe\/H] differences are not observed between first-\nand second-generation cluster stars. There are a few models that are\nexplicit about the importance of this coincidence, such as\nCarretta et al. (2010a)\\nocite{CBG10} and Smith (2010)\\nocite{S10}. Both of these models incorporate\nsupernova material into the second generation but require that it mix\nwell with accreted lower-metallicity gas before the formation of the\nsecond generation in order to make the metallicity of the second\nstellar generation be the same as the first. While these conditions\nare certainly conceivable for certain proto-globular clusters, they\nare unlikely to hold for all globular clusters in the Milky Way. \n\n\\section{Recent observational progress}\nThere have recently been several large-scale light-element abundance studies, which have the distinct advantage over smaller-scale studies that the abundance behavior of multiple clusters can be directly compared within homogeneous observations, data reduction and analysis. The low-\\linebreak resolution study of Kayser et al. (2008)\\nocite{KHG08} measured CN and CH bandstrengths in stars from the main sequence to the red giant branch in 8 Southern globular clusters. Those authors confirmed the presence of RGB CN bandstrength variations and CN-CH anticorrelations, as found in previous single- and few-cluster studies. They also demonstrated that CN bandstrength variations can be found on the main sequence in the clusters NGC 288, NGC 362, M22 and M55, clusters that had not previously been observed to contain main-sequence abundance variations. To explore the question of the source of light-element abundance variations, they also evaluate possible correlations between the ratio of CN-strong to CN-weak stars and several cluster parameters, and find mild positive correlations to cluster luminosity and tidal radius. These trends are interpreted as signs that globular clusters with larger masses or outer-halo orbits would be more efficient at producing second-generation stars.\n\nThe comprehensive study of Carretta et al. (2009b)\\nocite{CBG09} reported homogeneous oxygen\nand sodium abundances for 1958 stars of all evolutionary phases in 19\nSouthern globular clusters. This study made a significant statement\nabout the universality of light-element abundance variations in\nglobular clusters, and also explicitly adopted the language of\nself-enrichment and multiple stellar generations. By identifying\ngroups of stars as ``primordial'', ``intermediate'' or ``extreme''\ndepending on oxygen and sodium abundances, this study made the claim\nthat the degree of abundance variation can differ between clusters,\nand may be a function of environment and feedback\nsource. Main-sequence stars in many of the same clusters were observed\nby Pancino et al. (2010)\\nocite{PR10}, who measured CN and CH bandstrength variations from\nlow-resolution spectra. These authors found that the fraction of\nCN-strong (second-generation) stars was $\\sim 30\\%$, distinctly lower\nthan the $70\\%$ reported in Carretta et al. (2009b)\\nocite{CBG09}. This discrepancy is\ncurious, and Pancino et al. (2010)\\nocite{PR10} suggest that it may indicate that C-N\nabundance variations are contributed, at least in part, by a different\nfeedback source from the O-Na abundance variations studied by\nCarretta et al. (2009b)\\nocite{CBG09}. \n\nFrom recent large-scale studies it appears that light-\\linebreak element\nabundance variations are universal in Galactic globular clusters, but\nthe question of the dominant first-\\linebreak generation feedback source remains\nunsolved. Future studies will need to measure the C-N and O-Na variations\nsimultaneously in order to address the mismatch in frequency of\nsecond-generation stars found in\nlow- versus high-\\linebreak resolution spectroscopic studies, and will need to measure\nother specific elemental abundances to evaluate various aspects of\ncluster formation scenarios. For example, abundances of s-process\nelements like Ba would be useful for placing limits on AGB\ncontributions (e.g., Smith 2008\\nocite{S08};\\linebreak Yong et\nal. 2008\\nocite{YK08}), and the abundance of Li carries a great deal\nof information about the importance of infalling pristine gas (e.g.,\nD'Orazi \\& Marino 2010\\nocite{DM10}; Shen et al. 2010\\nocite{SB10}).\n\nThe APOGEE survey (Allende Prieto et al. 2008)\\nocite{AP08}, one of four\ncomponents of the SDSS-III project, will obtain\\linebreak high-resolution\nnear-infrared spectra for 100~000 stars in all components of the\nGalaxy, including red giants in globular clusters. The data reduction\npipeline will automatically determine 14 elemental abundances,\nincluding overall [Fe\/H] metallicity, $\\alpha$ elements, and most of\nthe light elements that vary in globular clusters. It will provide a\ndatabase for\\linebreak studying cluster light-element variations that is\nunparalleled in sample size, amount of abundance information per star,\nand start-to-finish homogeneity, and ought to shed significant light\non many aspects of cluster light-element abundance variations.\n\n\\section{Evolution of the Galactic globular cluster system}\nAlthough efforts have been made to understand the presence or degree\nof light-element abundance variations as a function of present-day\nglobular cluster properties, correlations with present-day mass,\nconcentration or ellipticity are loose at best. While we expect the\ntotal mass or central density of a cluster during the formation of the\nfirst and second generations to have an influence on its ability to\nself-enrich, those properties have clearly evolved significantly over\neach cluster's lifetime.\n\n\\subsection{Self-enrichment and escape velocity}\nOne of the more perplexing elements of the question of globular cluster\nlight-element self-enrichment is the fact\\linebreak that most of the Galactic\nglobular cluster population is unable to retain AGB or massive-star\nwinds at the present day. This is observable both in the lack of\nintracluster material\nin globular clusters (Evans et al. 2003\\nocite{ES03}; van Loon\net al. 2006\\nocite{vS06}; Boyer et al. 2006\\nocite{BW06}), and in low present-day escape\nvelocities. The census of Galactic globular clusters conducted by\nMcLaughlin \\& van der Marel (2005)\\nocite{Mv05} includes values for $v_{esc}$, and roughly half of the\nclusters have $v_{esc}$ below $20 {\\rm km s}^{-1}$. Since these clusters all have\nlight-element abundance variations, and since all proposed sources of\nfeedback material have wind speeds $\\ge 20 {\\rm km s}^{-1}$, these clusters must\nhave had higher escape velocities in the past. The massive first\ngeneration in the D'Ercole et al. (2008)\\nocite{DVD08} model provides one natural solution to\nthis problem, as do suggestions (e.g., Palou{\\v s} et al. 2009\\nocite{PW09}; Sills \\& Glebbeek 2010\\nocite{SG10}) that collisions between\nwinds from multiple stars should result in a lower bulk wind velocity,\ntrapping wind material that otherwise would escape in the dense inner regions of proto-globular\nclusters. It seems clear that the Galactic globular cluster population\nhas evolved strongly since its formation, both in terms of the overall\ncluster mass function (e.g., Parmentier \\& Gilmore 2005\\nocite{PG05}) and in the structural\nproperties of individual clusters (e.g., de Marchi et al. 2010\\nocite{dMP10}).\n\n\\subsection{The initial cluster mass function}\nIt is curious that light-element abundance variations\nare apparently universal among present-day Galactic globular\\linebreak clusters,\nconsidering that the initial cluster mass function included many\nlow-mass clusters that should not have been able to self-enrich according to\ncurrent globular cluster formation models. There are two possible\nexplanations for this coincidence that are quite simple: that\nself-enrichment in globular clusters is very common, and occurs at\nlower cluster masses than we expect, or that cluster dissolution was\nextremely effective early in the lifetime of the Milky Way, with only\na small percentage of the highest-mass clusters surviving to the\npresent day. Globular cluster formation scenarios that rely on\nsignificant gas infall\ntend to promote the first explanation, allowing clusters with\nlower-mass first generations to form a second stellar generation. The\nnumerical study of Marks \\& Kroups (2010)\\nocite{MK10} found that the expulsion of residual\ngas following star formation is very effective at destroying globular\nclusters with low initial masses and concentrations. This result both\nsupports the second explanation and implies that globular clusters\nhave contributed significant numbers of stars with first-generation\nabundances to the construction of the stellar halo of the Milky Way,\nas is also suggested by the result of Martell \\& Grebel (2010)\\nocite{MG10}.\n\nIf it is simply coincidental that the minimum mass for a globular\ncluster to survive to the present day in the Milky Way is larger than\nthe minimum mass for a globular cluster forming in the Milky Way to\nhost two stellar generations, then it is instructive to consider\nenvironments where those conditions are not met. In galactic\nenvironments that are more hospitable to long-lived low-mass globular\nclusters, the present-day cluster populations ought to include Milky\nWay-like, high-mass, two-population clusters along with lower-mass,\nchemically homogeneous globular clusters. In galactic environments\nin which it is more difficult for clusters to self-enrich, there would\nbe some fraction of high-mass globular clusters with homogeneous\nlight-element\\linebreak abundances. Regarding the first possibility, the\ntheoretical study of Conroy \\& Spergel (2011)\\nocite{CS10} suggests that inter-\\linebreak mediate-aged\nclusters in the Large Magellanic Cloud with masses between\n$10^{4}{\\rm M}_{\\odot}$ and $10^{5}{\\rm M}_{\\odot}$ should be able to retain\nfirst-generation winds and self-enrich because of the relatively low\nram pressure they experience. This claim is bolstered by the\nobservational study of Milone et al. (2009)\\nocite{MBP09}, in which clearly broad and\/or\nbifurcated main-sequence turn-\\linebreak offs were found to be common in\nintermediate-aged LMC clusters.\n\n\\section{Future challenges}\n\nIn order to correctly interpret the photometric complexities observed\nin some globular clusters (e.g., Marino et al. 2008\\nocite{MVP08}; Milone et\nal. 2008\\nocite{MBP08}; Han et al. 2009\\nocite{HL09}; Lardo et al. 2011\\nocite{LBP11}), we must understand the photometric shifts caused by changes in light-element abundances and helium, in addition to those caused by age, overall metallicity and [$\\alpha$\/Fe]. Current theoretical isochrones (e.g., Bertelli et al. 2008\\nocite{BG08}; Dotter et al. 2008\\nocite{DC08}; Han et al. 2009\\nocite{HK09}) are built from stellar models that allow variations in age, overall metallicity, and sometimes the abundances of $\\alpha$-elements and helium. Considering the correlations between light-element abundances and $U$-band photometry reported by, e.g., Marino et al. (2008)\\nocite{MVP08}, it seems prudent to expand the theoretical grid of stellar models to test for photometric sensitivity to light-element abundance variations. The study of Dotter et al. (2007)\\nocite{DCF07} considered exactly this question, constructing isochrones with enhancements in one of C, N, O, Ne, Mg, Si, S, Ca, Ti, or Fe while maintaining a constant overall heavy-element abundance Z in order to explore the effects of individual-element abundance variations on stellar structure. They find that enhancement in C, N or O abundance caused the isochrones to shift to the blue and reduced main-sequence lifetimes by as much as $15\\%$, while an enhanced Mg abundance caused iso-\\linebreak chrones to be redder but had a minimal positive effect on main-sequence lifetimes. They did not calculate isochrones for the anticorrelated light-element abundance pattern found in globular clusters, but such an exercise would be\\linebreak extremely helpful to our understanding of photometric complexity in globular clusters.\n\nIt will also be important to understand whether photometric variations are a generic result of light-element abundance variations, or if not, which globular cluster properties permit or prohibit them from being observed. As an example, large variations in CN and CH bandstrength are almost certainly responsible for $U$-band variations among red giants in relatively high-metallicity ([Fe\/H $\\ge -1.5$) globular clusters, but not all relatively high-metallicity clusters are known to have complex $U-B$,$B$ CMDs. Additionally, multiplicities in different regions of the CMD do not always correspond. For instance, the cluster $\\omega$ Cen has three main sequences and five distinct subgiant branches (Villanova et al. 2007)\\nocite{VPK07}, making it unclear how many distinct populations it contains. A search by Piotto (2009)\\nocite{P09} of archival HST\/ACS photometry uncovered multiple turnoffs in several clusters, and further searches for UV-blue photometry of globular cluster stars in public databases (as done in SDSS by Lardo et al. 2011\\nocite{LBP11}) or observatory archives could be a quick and profitable way to confirm or deny the presence of photometric complexity in a large number of Galactic globular clusters.\n\nDeveloping tools for interpreting integrated spectra of extragalactic globular clusters will dramatically expand our ability to study the effects of cosmological environment on globular cluster formation and self-enrichment. Methods for deriving ages and mean elemental abundances from low-resolution spectra have been adapted from galactic stellar populations studies (e.g., Puzia et al. 2006\\nocite{PKG06}; Schiavon 2007\\nocite{S07}), and techniques for extracting mean abundances from high-resolution integrated spectra of extragalactic globular clusters have been developed by Colucci et al. (2009)\\nocite{CB09}. A merger of the two approaches, matching high-resolution spectroscopic data to synthetic spectra that are a sum over multiple distinct populations, will allow detailed searches for abundance variations in extragalactic globular clusters to very large distances.\n\nIt is becoming an accepted paradigm that the majority of, if not all, ``normal'' Galactic globular clusters contain stars with a range of light-element abundances, although they are resolutely mono-metallic. This requires that clearly multi-metallic, multi-age clusters like $\\omega$ Cen and M54\\linebreak formed in different environments, and not as subsystems of the Milky Way. Rather, their extended star formation histories and ability to retain supernova feedback indicate that their early development occurred in a fairly high-mass environment. M54 lies quite close to the core of the Sagittarius dwarf galaxy, prompting some to claim that it formed as the nucleus of the galaxy (e.g., Layden \\& Sarajedini 2000\\nocite{LS00}), while others argue that M54 formed as a normal globular cluster but is being trapped in the galactic nucleus (e.g., Bellazzini et al. 2008\\nocite{BIC08}). One group (Carretta et al. 2010a)\\nocite{CBG10} has made the claim that M54 and $\\omega$ Cen both originated as nuclear star clusters in dwarf galaxies, with $\\omega$ Cen having been captured by the Milky Way earlier while M54 is still being removed from its galaxy of origin. The schematic model of multi-metallicity globular clusters having formed as nuclear star clusters in dwarf galaxies (e.g., Georgiev et al. 2009\\nocite{GHP09}) is attractive: the dark-matter halo of the galaxy would permit the cluster to experience extended feedback and star formation, and present-day nuclear star clusters are similar to multi-metallicity globular clusters in several properties such as half-light radius, escape velocity and horizontal branch morphology. \n\nRecent announcements of mild [Fe\/H] and [Ca\/Fe] variations in NGC 2419 (Cohen et al. 2010)\\nocite{CK10}, along with the discovery of photometric complexity (which may be a result of age or metallicity variations) in several otherwise unexceptional globular clusters (e.g., Piotto 2009\\nocite{P09}), raise the question of whether there is a class of globular clusters intermediate between ``normal'' mono-metallic, light element-variable globular clusters and the more massive multi-\\linebreak metallicity clusters. Theoretical studies of supernova feedback in extremely massive proto-globular clusters would help to clarify the feasibility of claiming that clusters with mild metallicity variations constitute the high-mass end of typical globular cluster self-enrichment. Numerical simulations of interactions between nucleated dwarf galaxies and the Milky Way would provide an estimate of how many nuclear star clusters may have been captured into Milky Way orbit, and whether clusters like NGC 2419 can be considered as examples of captured nuclear star clusters with a history of low-efficiency feedback. \n\n\\acknowledgements{SLM would like to thank the Scientific Organizing Committee of the Astronomische Gesellschaft 2010 Annual Meeting for the invitation to speak on this subject.}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nExplaining the origin of the current acceleration of the universe is\none of the biggest challenges in cosmology. Despite the great\nsuccesses of the $\\Lambda$CDM paradigm,  it is hard to accept that a\nvery fine-tuned form of otherwise invisible energy governs nowadays\nthe behavior of the universe at the largest scales. A whole plethora\nof models are proposed as alternatives to this situation. These are\nknown as \\emph{quintessence} or \\emph{dark energy} models\n\\cite{DEBook,Amendola:2012ys,Copeland:2006wr}. The motivations and\ndomain of applicability of various proposals are quite diverse:\nwhereas some of them are intended as mere phenomenological models only\nvalid for cosmology, others are rooted in theoretical considerations\nand are testable by different types of experiments. From the\ntheoretical viewpoint the preferred models are those addressing (at\nleast partially) the naturalness problem of the cosmological constant\nwithout introducing any additional fine-tunings.  In this paper we\nwill study in detail the  $\\Theta$CDM proposal of Ref.~\\cite{Blas:2011en}\nthat has such a property. This model is well-motivated theoretically\nand has a rich phenomenology that may distinguish it from\n$\\Lambda$CDM. \n\nThe $\\Theta$CDM model is based on the idea that the Lorentz invariance\nobserved in the Standard Model of particle physics is an emergent\nphenomenon and does not correspond to a symmetry of nature. This\nconcept is motivated by attempts to find complete theories of quantum\ngravity, but it can also be considered independently. There exist two\ndifferent approaches: on one hand Einstein-aether theory\n\\cite{Jacobson:2000xp,Jacobson:2008aj} provides a phenomenological\ndescription of gravity with broken Lorentz symmetry at large\ndistances; on the other hand Ho\\v rava gravity\n\\cite{Horava:2009uw,Blas:2009qj} invokes a violation of Lorentz\ninvariance at any scale to improve the quantum properties of\ngravitational theories.  The restriction of Ho\\v rava gravity to\noperators  with the lowest dimension can be considered as an effective\ntheory by itself, called the \\emph{khronometric} theory\n\\cite{Blas:2010hb}.  The two proposals are very similar at large\ndistances: the khronometric theory can be viewed as a constrained\nversion of the Einstein-aether theory, and in many situations the\npredictions of the two theories coincide\n\\cite{Jacobson:2010mx,Blas:2010hb}.  The analysis presented in this\nwork is applicable to all these classes of models.  However, in the\nlast step consisting in deriving the actual observational bounds on\nthe model parameters, we will restrict to the khronometric case. \n \nSeveral implications of the above models for cosmology have been\nstudied in the past,  see e.g.\n\\cite{Zuntz:2008zz,Kobayashi:2010eh,ArmendarizPicon:2010rs,\nCarroll:2004ai,Nakashima:2011fu,Li:2007vz,Zlosnik:2007bu}.  For the\nevolution of the Universe as a whole, the consequences of the minimal\nmodels\\footnote{By this term we refer to the models with a minimal\n  number of additional degrees of freedom in the gravity sector, and\n  the simplest kinetic action (Einstein-aether and khronometric\n  theories) \\cite{Jacobson:2008aj,Blas:2010hb}.} \nare rather trivial. They are compatible with\nFriedmann--Robertson--Walker (FRW) solutions that differ from the\nstandard case only by a renormalization of the gravitational constant\naway from the value measured in local tests of Newton's law.  In other\nwords, those models do not modify the form of the Friedmann equation.\nThe background evolution becomes more interesting  when non-minimal\nmodels are considered. In this case, the breaking of Lorentz\ninvariance allows for Lagrangians which can change the expansion\nhistory of the universe and provide new alternatives for inflationary\ndynamics, dark matter and dark energy\n\\cite{Donnelly:2010cr,Blas:2011en,Zlosnik:2007bu,Zuntz:2008zz}. Some\ncriteria are necessary to identify the most interesting cases.  In\nthis work we will be concerned with the issue of dark energy, for\nwhich one would like to find a simple model described by a Lagrangian\nwith high cutoff scale, that provides a mechanism to accelerate the\nuniverse insensitive to UV  corrections\\footnote{We leave aside the\n  ``old cosmological constant problem'': to find a mechanism imposing\n  a null vacuum energy (see however the related comments in\n  \\cite{Blas:2011en}.)} \nand distinguishable from $\\Lambda$CDM. The $\\Theta$CDM model of\nRef.~\\cite{Blas:2011en} meets these requirements.\n\nSome consequences of $\\Theta$CDM have been already discussed in\nRef.~\\cite{Blas:2011en}.  In the present work we describe the\nobservable physical effects of the model on cosmic microwave\nbackground (CMB)  anisotropies and on the matter power spectrum (at\nthe linear level).  We provide a detailed discussion of such effects,\nthat we computed accurately with a modified version of the flexible\nBoltzmann code {\\sc class} \\cite{Blas:2011rf}. We then compare the\n$\\Theta$CDM model to recent CMB and Large Scale Structure (LSS) data using\nthe Monte Carlo parameter inference code {\\sc Monte Python}\n\\cite{Audren:2012wb}.\n \nOur work is organized as follows: in Sec.~\\ref{sec:Lagr} we describe\nthe $\\Theta$CDM model and discuss the constraints not related to\ncosmology. In Sec.~\\ref{sec:cosmo} we study the background evolution\nand derive linear equations for perturbations around the FRW\nbackground. The results presented in this section are complementary\nto those in Ref.~\\cite{Blas:2011en}. They hold in a different gauge,\nand refer to different conventions and parametrizations, found to be\nmore suitable for the numerical implementation.  The qualitative\neffects of $\\Theta$CDM on the CMB and matter power spectrum are described\nin Sec.~\\ref{sec:obser}. We present constraints from CMB and LSS data\nin Sec.~\\ref{sec:data}, and expose our conclusions in\nSec.~\\ref{sec:concl}. Appendix~\\ref{sec:app} contains the derivation\nof initial conditions for cosmological perturbations in $\\Theta$CDM.\n\n\\section{Lorentz breaking theories of gravity and $\\Theta$CDM}\\label{sec:Lagr}\n\n\nLorentz invariance is one of the best tested symmetries of the\nStandard model of particle physics \\cite{Kostelecky:2008ts}.  It is\nalso a fundamental ingredient of the theory of general relativity,\nthat provides a very successful description of gravitational\ninteractions over a huge range of scales. However, it is not known\nhow to directly promote general relativity to a complete quantum\ntheory, which points towards the necessity to consider alternatives.\nIndependently of this, modifications to general relativity are\ncurrently being considered in the area of cosmology. The rationale\nbehind these modifications is the possibility to use the wealth of\ncosmological data to learn how gravitation behaves at the largest\naccessible distances and, hopefully, shed some light on the mechanism\nresponsible for the accelerated expansion of the universe.\n\nThese two lines of research converge if one assumes that Lorentz\ninvariance is not a symmetry of the gravitational sector. This idea\nopens the possibility to construct gravitational theories with better\nquantum behavior than general relativity \\cite{Horava:2009uw}.  This\nis achieved at the price of introducing new degrees of freedom that\nmodify the laws of gravity at all distances, including those relevant\nfor cosmology \\cite{Blas:2009qj,Blas:2010hb}. Even before this\ntop-down approach had been initiated, the bottom-up Einstein-aether\nmodel \\cite{Jacobson:2000xp} was proposed as a way to capture the\nlarge-distance effects of a putative Lorentz violation due to quantum\ngravity.  In this work, we will consider theories where Lorentz\nviolation is described by a preferred time-like vector field $u_\\m$\ndefined at every point of space-time, which includes Einstein-aether\ntheory and Ho\\v rava gravity. For the latter,  we will focus only on\nits low-energy form, the khronometric theory \\cite{Blas:2010hb}.  The\nvector field will be normalized to\\footnote{We use the $(-+++)$\n  signature for the metric. This differs from most of the previous\n  works in the field, but is common in cosmology.}\n\\begin{equation}\n\\label{constr}\nu_\\m u^\\m=-1.\n\\end{equation}\nThe presence of this dynamical field allows to use for the description\nof Lorentz violation the same language as for spontaneous symmetry\nbreaking.  Gravitational physics at large distances is governed by the\nfollowing covariant action for $g_{\\m\\n}$ and $u_\\m$, featuring a\nminimal number of derivatives:\n\\begin{equation}\n\\begin{split}\n\\label{aetheract}\nS_{[ \\mathrm{EH}u]}=\\frac{M_0^2}{2}\\int \\mathrm d^4x \\sqrt{-g}\n\\large[R\\,-\\,&K^{\\m\\n}_{\\phantom{\\m\\n}\\s\\r}\\nabla_\\m u^\\s\\nabla_\\nu u^\\rho\\\\\n&\\quad\\quad+l (u_\\m u^\\m+1)\\large]\\;,\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{Kmnsr}\nK^{\\m\\n}_{~~~\\s\\r}\\equiv c_1g^{\\m\\n}g_{\\s\\r}+c_2\\delta_\\s^\\m\\delta_\\r^\\n\n+c_3\\delta_\\r^\\m\\delta_\\s^\\n-c_4u^\\m u^\\n g_{\\s\\r},\n\\end{equation}\nand $l$ is a Lagrange multiplier that enforces the unit-norm\nconstraint. Equation~(\\ref{aetheract}) includes the Einstein-Hilbert\n(EH) term evaluated with the metric $g_{\\m\\n}$. The parameter $M_0$ is\nproportional to the Planck mass (cf. \\eqref{gn}) while the\ndimensionless constants $c_a$, $a=1,2,3,4$, characterize the strength\nof the interaction of the aether  $u_\\m$ with gravity. This is the\naction of the Einstein-aether model\n\\cite{Jacobson:2000xp,Jacobson:2008aj}.\n\nThe khronometric case corresponds to the situation where $u_\\m$ is\ndefined as a vector normal to a foliation consisting of the level\nsurfaces of the {\\it khronon} field $\\varphi$, \n\\begin{equation}\n\\label{ukh}\nu_\\m\\equiv\\frac{\\partial_\\m \\varphi}{\\sqrt{-\\nabla^\\n \\varphi \\partial_\\n \\varphi}}.\n\\end{equation}\nIn this case, the constraint (\\ref{constr}) is satisfied identically\nand the first term of (\\ref{Kmnsr}) can be expressed as a linear\ncombination of the last two terms. Thus, $K^{\\m\\n}_{~~~\\s\\r}$\nreduces to its last three terms with coefficients \n\\begin{equation}\n\\label{khpar}\n\\l\\equiv c_2,~~~\\beta\\equiv c_3+c_1,~~~\\alpha\\equiv c_4+c_1\\;.\n\\end{equation}\nAt the level of linear perturbations, the khronometric theory differs\nfrom the Einstein-aether theory by the number of propagating degrees\nof freedom apart from the spin-two mode of the graviton. While the\naether in general describes vector and scalar excitations, the\nkhronometric case only contains scalars. However, the scalar sectors\nof the two theories are equivalent and are completely characterized by\nthe three parameters (\\ref{khpar}). \n \nOnce the couplings to matter are specified, the constants $c_a$ are\nconstrained by various considerations ranging from theoretical\nrequirements to observational tests. First, to very good precision,\nthe Standard Model fields must couple only to $g_{\\m\\n}$ and not to\n$u_\\m$, as required by Lorentz invariance in this sector\\footnote{More\n  generally, a universal coupling of Standard Model fields to a fixed\n  combination of $g_{\\m\\n}$ and $u_\\m$ is allowed. This reduces to the\n  previous case by a redefinition of the metric \\cite{Foster:2005ec}.} \n\\cite{Kostelecky:2008ts}.  This decoupling presents a serious\nchallenge to the proposal, but it is conceivable to achieve it either\nby imposing extra symmetries, e.g. supersymmetry\n\\cite{GrootNibbelink:2004za,Pujolas:2011sk}, or through a\nrenormalization group running \\cite{PS2}.  Next, there are\nrestrictions imposed by the stability of Minkowski spacetime\n\\cite{Jacobson:2008aj}. In particular, the requirement that the scalar\nmode is neither a ghost nor a tachyon field amounts to the constraints\n\\begin{equation}\n\\label{stab}\n0<\\a<2~,~~~\\b+\\l>0\\;.\n\\end{equation}\nStringent bounds come from observations of the Solar System dynamics,\nwhich can be used to place constraints on post-Newtonian (PPN)\nparameters. Two of these parameters, denoted by $\\a_1^{PPN}$ and\n$\\a_2^{PPN}$, describe the effects of Lorentz violation. In general,\nthey are different from zero (their value in general relativity) both\nin the Einstein-aether and khronometric theory; we refer the reader to\n\\cite{Jacobson:2008aj,Blas:2011zd} for explicit relations with the\nconstants $c_a$. Assuming no cancellations in these formulae, one\nobtains experimental bounds of the order of\n\\begin{equation}\n\\label{eq:bounda}\n|c_{a}|\\lesssim 10^{-7}\\;.\n\\end{equation}\nHowever, there are regions in the space of model parameters $(\\alpha$,\n$\\beta$, $\\lambda$) in which $\\a_{1,2}^{PPN}$ vanish and Solar System\ntests are automatically satisfied, with Newton's constant given by \n\\begin{equation}\n\\label{gn}\nG_{N}\\equiv\\frac{1}{8\\pi M_0^2(1-\\alpha\/2)}\\;.\n\\end{equation}\nThis requires\\footnote{The difference between the models is due to the\n  contributions to $\\a_{1,2}^{PPN}$ from the vector polarizations\n  that are present only in the Einstein-aether case.}\n\\begin{subequations}\n\\label{abl}\n\\begin{align}\n\\label{abkhronon}\n&\\a=2\\b~~ \\text{for the khronometric model,}\n\\\\\n\\label{ablaether}\n&\\alpha=-(3\\lambda+\\beta)~~ \\text{for the Einstein-aether.}\n\\end{align} \n\\end{subequations}\nIn these cases, much weaker bounds can be inferred from gravitational\nwave emission in binary systems \\cite{Foster:2006az,Blas:2011zd},\ngiving\n\\begin{equation}\n\\label{eq:boundb}\n|c_{a}|\\lesssim 10^{-2}\\;,\n\\end{equation}\nand from the form of black hole solutions, imposing inequalities\ndetailed in~\\cite{Barausse:2011pu}.  We will see that in the case of\nthe khronometric model, the bounds that can be inferred from cosmology\nare competitive with (\\ref{eq:boundb}), but not with\n(\\ref{eq:bounda}). Thus we will impose the relation (\\ref{abkhronon})\nwhen searching for the allowed parameter space.  Previous studies of\nthe cosmological effects of Lorentz violation\n\\cite{Zuntz:2008zz,Li:2007vz} focused on the Einstein-aether model.\nIn those works, the relation (\\ref{ablaether}) was enforced to avoid\nthe PPN constraints. As we will explain, the relation\n(\\ref{ablaether}) also incidentally suppresses the leading effects in\ncosmology, unlike the relation (\\ref{abkhronon}). This explains why\nthe bounds obtained in those studies are rather mild. \n\nThe implications of the action (\\ref{aetheract}) for the background\nevolution of an homogeneous and isotropic universe are minimal. If all\nmatter components are universally coupled\\footnote{A priori, there is\n  no reason why this must be true for the dark matter. Still, as shown\n  in \\cite{Blas:2012vn}, even allowing for non-universal interactions\n  between the dark matter and the aether does not change the\n  background evolution. As our main focus in this paper is dark\n  energy, we will assume that the dark matter has standard properties\n  (namely, that it is a pressureless fluid, described at the\n  fundamental level by a Lorentz invariant Lagrangian, and with\n  universal coupling to $g_{\\m\\n}$).}\nto $g_{\\m\\n}$, the only difference  with respect to general relativity\nis that the Friedmann equation involves a renormalized gravitational\nconstant\n\\begin{equation}\n\\label{gc}\nG_{cos}\\equiv\\frac{1}{8\\pi M_0^2(1+\\beta\/2+3\\lambda\/2)}\\;\n\\end{equation}\n differing from the value $G_N$ measured e.g. on earth or in the Solar\n System\\footnote{The analysis of Big Bang Nucleosynthesis\n   \\cite{Carroll:2004ai} sets a bound on the relative difference\n   between the two, $|G_{cos}\/G_N-1|\\leq 0.13$.},\ngiven by Eq.~(\\ref{gn}).  In order to modify the expansion history of\nthe universe, and find a candidate for dark energy, we need to add a\nnew ingredient to the model. We want to do it in a way that preserves\nthe simplicity of the proposal and its validity as a low-energy\neffective field theory.  This is achieved by supplementing the action\n(\\ref{aetheract}) with a new field $\\Theta$ invariant under the shift\nsymmetry,\n\\begin{equation}\n\\label{eq:shift}\n\\Theta\\mapsto \\Theta+const.\n\\end{equation}\nThe low-energy action for this field is\\footnote{ A similar action\n  with an additional potential term for $\\Theta$ breaking the shift\n  symmetry (\\ref{eq:shift}) was considered in \\cite{Donnelly:2010cr}.}\n\\begin{equation}\\begin{split}\n\\label{Thetaact}\nS_{[\\Theta]}=\\int \\mathrm d^4x \\sqrt{-g}& \\bigg(-\\frac{g^{\\m\\n}\\partial_\\m\\Theta\\partial_\\n\\Theta}{2}\\\\\n~~~~~&+\n\\kappa\\frac{(u^\\mu\\partial_\\mu\\Theta)^2}{2}\n-\\mu^2u^\\mu\\partial_\\mu\\Theta\\bigg)\\;,\n\\end{split}\n\\end{equation}\nand involves two free parameters ($\\kappa$, $\\mu$).  We will refer to\nthe model resulting from the combination of the actions\n(\\ref{Thetaact}) and (\\ref{aetheract}) (with a universal coupling\nbetween matter fields and $g_{\\m\\n}$) as $\\Theta$CDM \\cite{Blas:2011en}.\nThe cosmological constant term is set to be exactly zero\\footnote{One\n  can entertain the possibility of finding a mechanism that would\n  enforce the cancellation of the vacuum energy induced by quantum\n  loops \\cite{Blas:2011en}. However, at present, we are not aware of\n  any such mechanism. Thus, the vanishing of the cosmological constant\n  should be taken merely as an assumption.}. \nTwo comments are in order. First, as an effective field theory,\n$\\Theta$CDM provides a valid description of physics up to  a cutoff scale\nof the order of $\\Lambda_c\\sim \\sqrt{c_a}M_P$.  The bounds\n(\\ref{eq:bounda}) or (\\ref{eq:boundb}) show that this cutoff scale may\nbe only a few orders of magnitude below the Planck mass.  Furthermore,\nin the khronometric case, the theory has a potential UV completion,\nsince it is a sub-case of Ho\\v rava gravity \\cite{Horava:2009uw}.\nSecond, in the whole $\\Theta$CDM action, only the last operator in\n(\\ref{Thetaact}) breaks the discrete symmetry $\\Theta\\to -\\Theta$. This\nimplies that from the viewpoint of the effective field theory, it is\nself-consistent to choose the coefficient in front of this operator to\nbe much smaller than the UV cutoff. In spite of $\\m$ being\ndimensionfull, the above symmetry guarantees that it is renormalized\nmultiplicatively, and that no dangerous contributions proportional to\n$\\Lambda_c$ appear. In other words, the smallness of $\\m$ is\ntechnically natural.  This last observation is very important, since\nwe are going to see that in the $\\Theta$CDM  model, $\\mu$ sets the scale\nof the current cosmic acceleration.\n\n\n\\section{Cosmological solutions of $\\Theta$CDM}\\label{sec:cosmo}\n\nWe are interested in describing the evolution of perturbations around\nhomogeneous and isotropic solutions in the $\\Theta$CDM model.  We focus on\nscalar perturbations, and refer the reader to\n\\cite{Nakashima:2011fu,ArmendarizPicon:2010rs,Lim:2004js} for possible\neffects of vector and tensor modes. In the synchronous gauge, the\nperturbed FLRW metric has the form, \n\\begin{equation}\n\\label{FRW}\n\\begin{split}\n&g_{00}=-a(\\tau)^2, \\quad g_{0i}=0, \\quad\\\\\n&g_{ij}=a(\\tau)^2\\left[\\delta_{ij}+\\frac{\\partial_i\\partial_j}{\\Delta} h+6\\left(\\frac{\\partial_i\\partial_j}{\\Delta}-\\frac{1}{3}\\delta_{ij}\\right)\\eta\\right].\n\\end{split}\n\\end{equation}\nBesides, for $\\Theta$ and the khronon (or the longitudinal component\nof $u_\\m$ in the more general Einstein-aether case) we introduce \n\\begin{equation}\n\\Theta=\\bar \\Theta(\\tau)+\\xi, \\quad \\varphi=\\tau+\\chi.\n\\end{equation}\nThe matter components are assumed to be cold dark matter ($cdm$),\nphotons ($\\gamma$), neutrinos ($\\n$) and baryons ($b$).  We describe\nthese components at the same level of approximation as in\nRef.~\\cite{Ma:1995ey}.  In particular, the dark matter is treated as a\npressureless fluid universally coupled to the metric $g_{\\m\\n}$.  We\nassume that it is comoving with the gauge, in order to eliminate the\nwell-known residual freedom in the synchronous gauge.  We now discuss\nhow the gravitational equations are modified in $\\Theta$CDM.  These\nequations were derived in \\cite{Blas:2011en} using the conformal\nNewtonian gauge.  Here we will rewrite the linearized equations in the\nsynchronous gauge \\eqref{FRW}, and in a form optimized for numerical\nstudy with the Boltzmann code {\\sc class} \\cite{Blas:2011rf}. \n\n\\subsection{Background evolution}\n\nDeriving the equation of motion for $\\Theta$ from (\\ref{Thetaact}), one\nfinds that the homogeneous part $\\bar\\Theta(t)$ evolves as\n\\begin{equation}\n\\label{eq:backg}\n\\dot{\\bar \\Theta}=-\\frac{\\m^2  a}{1+\\kappa}+\\frac{C}{a^2}\\;,\n\\end{equation}\nwhere $C$ is an integration constant. Substituting this solution into\nthe Friedmann equation (derived from the combination of the actions\n(\\ref{aetheract}) and (\\ref{Thetaact})) yields\n\\begin{equation}\n\\label{eq:Friedm}\nH^2\n=\\frac{8\\pi G_{cos}}{3}\\left(\\rho_\\m+\\rho_s+\\rho_d+\\sum_{\\rm other}\\rho_n\\right),\n\\end{equation}\nwhere $H\\equiv \\dot a\/a^2$ and $G_{cos}$ is given by (\\ref{gc}). The\nfirst three contributions in the brackets come from the\nenergy-momentum tensor of the $\\Theta$-field and have the form, \n\\begin{equation}\n\\rho_\\m\\equiv\\frac{\\m^4}{2(1+\\kappa)}\\;, \n\\quad \\rho_s\\equiv\\frac{C^2(1+\\kappa)}{2a^6}\\;,\\quad\n\\rho_d=-\\frac{\\m^2C}{a^3}\\;,\n\\end{equation}\nwhile $\\r_n$, $n=cdm,\\gamma,\\n,b$, stand for the densities of the\nstandard matter components of the Universe.\n\nLet us analyze Eq.~(\\ref{eq:Friedm}). Recall that there is no\ncosmological constant at the fundamental level in the model. Instead,\nthe first term in (\\ref{eq:Friedm}) plays the same role with an energy\nscale set\\footnote{Throughout the paper we assume that the combination\n  $1+\\kappa$ is of order one.}\nby $\\mu$.  As emphasized above, $\\m$ does not receive large radiative\ncorrections, and thus this source of dark energy can naturally have an\nenergy scale completely unrelated to the cutoff of the theory.  The\nsecond term has the form of the contribution of stiff matter.  Not to\nspoil Big Bang Nucleosynthesis (BBN), $\\rho_s$ must be smaller than\nabout 30 times the density of radiation  in the Universe at\ntemperatures of the order of 10 MeV \\cite{Dutta:2010cu}.  Thus,\nassuming that this contribution was already present at BBN\\footnote{We \n  do not specify the origin of the field $\\Theta$ in this work. One\n  possibility is to identify $\\Theta$ with the Goldstone boson of a\n  spontaneously broken global symmetry.  Then, the above assumption\n  amounts to stating that the corresponding phase transition occurs\n  before BBN. If it happens later, the picture may change, but it is\n  hard to see how this option can be incorporated in a viable\n  cosmological scenario.},\ndue to its rapid decrease, it is completely negligible at later\nepochs.  The third term in (\\ref{eq:Friedm}) behaves as the energy\ndensity of dust\\footnote{Note though, that its sign can be negative\n  depending on the sign of $C$.},\nso one might be tempted to identify it with the dark matter. However,\nbeing the geometric mean of the first two, this term is always\nsubdominant and cannot contribute a significant fraction of dark\nmatter. Given these considerations, we will set $C=0$ henceforth.\n\n\n \\subsection{Cosmological perturbations\\label{ssec:pk}}\n\\label{ssec:cosper}\n\nLet us introduce two time scales that appear in the analysis of the\nlinear perturbations,\n\\begin{equation}\n\\label{scales}\n\\tau^{-1}_\\a\\equiv\\sqrt\\frac{8\\pi G_{cos}}{\\a}\\,\\dot{\\bar\\Theta}\\;,\\quad\nH_\\a\\equiv\\frac{H_0}{\\sqrt{\\a}}\\;,\n\\end{equation}\nwhere $H_0 = 100\\ \\mathrm{h \\ km\\ s^{-1}\\ Mpc^{-1}}$ is the current\nvalue of the Hubble constant today.  It is also convenient to rescale\nthe $\\Theta$-fluctuation defining\n\\begin{equation}\n\\label{tildee}\n\\tilde \\xi  \\equiv \\frac{\\sqrt{8 \\pi G_{\\text{cos}}}}{H_0} \\xi\\;,\n\\end{equation}\nso that both fields $(\\chi,\\tilde  \\xi)$ have the dimension of time.\nThe equations of motion for $\\chi$ and $\\tilde \\xi$ read\n\\begin{subequations}\n\\label{eqchixi}\n\\begin{align}\n  \\ddot{\\chi} =& - 2 {\\cal H} \\dot{\\chi} \\nonumber\\\\\n  -\\bigg[&k^2c_\\chi^2 +  (1+B){\\cal H}^2 +(1-B)\\dot{\\cal H} +\n  \\frac{G_0}{G_{\\text{cos}} \\tau^{2}_\\a}    \\bigg] \\chi \\nonumber \\\\\n  &+\\frac{G_0}{G_{\\text{cos}}}\\frac{H_\\alpha}{\\tau_\\a} \\tilde\\xi \n-\\frac{c_\\chi^2}{2}\\dot h - 2\\frac{\\beta}{\\alpha}\\dot \\eta,\n  \\label{eq:chi}\\\\\n\\label{eq:xi}\n  \\ddot{\\tilde \\xi} =& -2\\mathcal{H} \\dot{ \\tilde \\xi} -\n  k^2c_\\Theta^2\\tilde{\\xi}+\\frac{k^2c_\\Theta^2}{H_\\a\\tau_\\a}\\chi, \n\\end{align}\n\\end{subequations}\nwhere we introduced the notations \n\\begin{equation}\n\\begin{split}\n& G_0 \\equiv \\frac{1}{8 \\pi M_0^2}\\;, \\quad\n\\frac{G_{cos}}{G_0} \\equiv\\frac{1}{1 + \\beta\/2 + 3 \\lambda\/2}\\;, \\\\\n&B\\equiv\\frac{\\beta+3\\lambda}{\\alpha}\\;,\\quad\nc_\\chi^2\\equiv \\frac{\\beta+\\lambda}{\\alpha}\\;,\n\\quad c^2_\\Theta\\equiv \\frac{1}{1+\\kappa}\\;,\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n{\\cal H}\\equiv a H.\n\\end{equation}\nNote that the constants $c_\\chi$, $c_\\Theta$ represent the sound speeds of the\nfields $\\chi$ and $\\xi$ at relatively short wavelengths, where these modes\ndecouple from each other \\cite{Blas:2011en}.  These two velocities are constant\nin time and we will treat them as quantities of order one, which is their\nnatural order of magnitude from the point of view of effective field theory.\nNote that in general they can exceed one, so that the fields can be\nsuperluminal. In Lorentz violating theories, this does not lead to any causal\nparadoxes, see e.g. the discussion in Ref.~\\cite{Mattingly:2005re}.\n\nWe now present the linearized Einstein equations. There are four\nequations in the scalar sector corresponding to different components\nof the Einstein tensor \n$G^\\m_{\\phantom{\\m}\\n}\\equiv R^\\m_{\\phantom{\\m}\\n}-\\frac{1}{2}\n\\delta^\\m_{\\phantom{\\m}\\n}R$.\nOnly two of these equations are independent, but we write all of them\nfor completeness. For the $\\delta G^0_{\\phantom{0}0}$ component we\nfind\n\\begin{align}\n  \\bigg(k^2\\eta - \\frac{1}{2}\\frac{G_0}{G_{\\text{cos}}}\\mathcal H \\dot h\\bigg) \n= -4\\pi a^2 {G_0}\\sum_n \\rho_n\\delta_n,\n  \\label{eq:T00}\n\\end{align}\nwith\n\\begin{equation}\n\\begin{split}\n \\sum_i \\rho_i\\delta_i \\equiv& \\sum_{\\rm other} \\rho_i\\delta_i +\n \\frac{\\a}{8\\pi a^2 G_{cos}c_\\Theta^2}\\frac{H_\\a}{\\tau_\\a}\\dot{\\tilde\n \\xi}\\\\\n  &+ \\frac{\\alpha\\, k^2}{8\\pi a^2G_0}  (\\mathcal H (1-B)\\chi + \\dot\\chi ).\n  \\label{}\n\\end{split}\n\\end{equation}\nHere and in what follows, the label `other' refers to contributions\nfrom the standard matter components, whose form can be found in\n\\cite{Ma:1995ey}. For the $\\partial_i\\delta G^0_{\\phantom{0}i}$ part we\nfind \n\\begin{align}\n  2k^2(1-\\beta)\\dot\\eta - \\frac{\\a\\,  c_\\chi^2}{2}k^2\\dot h = 8 \\pi a^2{G_0} \\sum_n(\\rho_n +p_n)\\theta_n, \n  \\label{eq:etadot}\n\\end{align}\nwith\n\\begin{equation}\n\\label{eq:theta}\n\\begin{split}\n\\sum_n(\\rho_n + p_n)\\theta_n \\equiv\\sum_{\\rm other}(\\rho_n +  p_n)\\theta_n +  \\frac{\\a\\,  c_\\chi^2}{8\\pi a^2G_0}k^4\\chi.\n\\end{split}\n\\end{equation}\nThe $\\delta G^i_{\\phantom{i}i}$ equation reads\n\\begin{align}\n\\label{eq:Tii}\n  \\ddot h &= -2\\mathcal H \\dot h + 2\\frac{G_{\\text{cos}}}{G_0} k^2 \\eta-24\\pi G_{cos}a^2\\sum_i\\delta p_n,\n\\end{align}\nwhere\n\\begin{equation}\n\\label{eq:press}\n\\sum_n\\delta p_n\\equiv \\sum_{\\rm other}\\delta p_n +  \\frac{\\a\\,\n  B}{24\\pi a^2G_0}k^2(\\dot \\chi + 2\\mathcal H\\chi)\\;. \n\\end{equation}\nFinally, the  $\\partial_i\\partial_j\\delta G^i_{j}$ equation yields\n\\begin{align}\n\\label{eq:tracefree}\n  (1-\\beta)\\big(\\ddot h + 6\\ddot \\eta + 2\\mathcal H (\\dot h +\n  6\\dot\\eta)\\big) \n&- 2k^2\\eta =\\nonumber\\\\-24\\pi a^2{G_0}\n  &\\sum_n( \\rho_n +  p_n)\\sigma_n,\n\\end{align}\nwith\n\\begin{equation}\n\\sum_n(\\rho_n + p_n)\\sigma_n \\equiv\\sum_{\\mathrm{other}}( \\rho_n + p_n)\\sigma_n  - \\frac{\\beta k^2}{12\\pi a^2G_0}(\\dot\\chi + 2\\mathcal H \\chi).\n\\end{equation}\nIn the above expressions the pressure fluctuations $\\delta p_n$, the\nvelocity divergences $\\theta_n$ and the shear potentials $\\sigma_n$\nare defined in the standard way, see \\cite{Ma:1995ey}; the equations\nfor perturbations in the matter components can be found in the same\nreference. The linearized equations must be supplemented  by suitable\ninitial conditions. The latter are derived in Appendix~\\ref{sec:app}.   \n\nFor simplicity, we are going to set $c_\\Theta=1$ in the numerical\nsimulations. This prescription is equivalent to fixing $\\kappa=0$, and\nleaves us with four free fundamental parameters ($\\alpha$, $\\beta$,\n$\\lambda$, $\\mu$).  A different choice for $c_\\Theta$ would not affect\nqualitatively the evolution of perturbations, unless $c_\\Theta$ is very\nsmall. In that case, the field $\\tilde{\\xi}$ could in principle\ncluster on scales much smaller than the Hubble radius, but we will not\nconsider this situation in the present work\\footnote{A small value of\n  $c_\\Theta$ corresponds to the near cancellation of the kinetic terms\n  for $\\tilde{\\xi}$ coming from the first and second terms in\n  Eq.~(\\ref{Thetaact}) - a situation that appears fine-tuned from the\n  effective field theory perspective.}.\n\nWe can compare the number of free parameters in this model with that\nin an ordinary $\\Lambda$CDM model sharing the same background\nevolution. The parameters ($\\Omega_\\Lambda$, $H_0$) of $\\Lambda$CDM\ncan be mapped onto the parameters ($\\mu$, $H_0$) of the $\\Theta$CDM model.\nHence the latter features only three additional parameters ($\\alpha$,\n$\\beta$, $\\lambda$), reducing to only two independent parameters after\nimposing one of the conditions fulfilling Solar System tests,\n(\\ref{ablaether}) or (\\ref{abkhronon}). Once ($\\alpha$, $\\beta$,\n$\\lambda$) are fixed, all coefficients in the field equations\n(\\ref{eqchixi}) and in the Einstein equations can be derived.\n\n\n\\subsection{Changing the gauge}\n\n\nReference \\cite{Ma:1995ey} shows the impact of a gauge transformation\non the variables describing matter  and metric perturbations, and on\nthe form of their respective equations of evolution.  Here we show how\nthe fields $\\chi$ and $\\tilde{\\xi}$ transform when switching from the\nsynchronous to the Newtonian gauge. This transformation is induced by\nthe particular change of coordinates\n\\begin{equation}\nx^0\\mapsto x^0-\\dot\\beta(x,\\tau)\\;, \n\\quad x^i\\mapsto x^i-\\partial_i \\beta(x,\\tau)-\\epsilon^i(x)\\;,\n\\end{equation}\nwhere\n\\begin{equation}\n\\beta(x,\\tau)=\\int \\mathrm d^3 k \\frac{e^{i kx}}{2k^2}(h+6\\eta)\\;,\n\\quad\n\\partial_i\\epsilon^i=0\\;.\n\\end{equation}\nAfter this transformation, the scalar part of the metric (\\ref{FRW})\nbecomes diagonal,\n\\begin{equation}\n\\label{Newmetric}\n\\mathrm d s^2=a(\\tau)^2\\left[-(1+2\\psi)\\mathrm d \\tau^2+(1-2\\phi)\\mathrm d x^i \\mathrm d x_i\\right],\n\\end{equation}\nwith\n\\begin{equation}\n\\psi=\\ddot\\beta+\\H\\dot\\beta\\;,\\quad\\phi=\\eta-\\H\\dot\\beta\\;.\n\\end{equation}\nThe khronon and $\\Theta$ field transform as\n\\begin{equation}\n\\chi\\mapsto\\chi^N=\\chi+\\dot\\beta\\;,\\quad\n\\tilde\\xi\\mapsto {\\tilde\\xi}^{N}=\n\\tilde\\xi+\\frac{\\sqrt{8\\pi G_{cos}}}{H_0}\\dot{\\Bar\\Theta}\\dot\\beta\\;.\n\\end{equation}\nAlthough the Boltzmann code {\\sc class} features equations in both the\nsynchronous and Newtonian gauge, for simplicity, we choose to\nimplement the $\\Theta$CDM equations in the synchronous gauge only.\nHowever, when presenting the physical interpretation of numerical\nresults in the next section, we will refer to the evolution of\nquantities in the Newtonian gauge, obtained by performing the above\ntransformation inside the code a posteriori (i.e. after solving the\nequations of motion in the synchronous gauge).\n\n\n\\section{Observable effects}\\label{sec:obser}\n\nThe modified evolution of linear perturbations in $\\Theta$CDM leads to\nobservable consequences which we now discuss. We focus on the scalar\nsector, known to contribute most to observable quantities.  The\neffects of Lorentz violation on the tensor and vector sectors are\nweakly constrained by current cosmological data\n\\cite{Nakashima:2011fu,ArmendarizPicon:2010rs,Lim:2004js}.  The only\neffect on tensor modes is a small shift in their velocity\n\\cite{Jacobson:2008aj}.  The vector equations are identical in general\nrelativity and in the khronometric model, implying that vector\nperturbations decay with time. Possible effects of vector modes in the\nEinstein-aether model on CMB polarization have been discussed in\n\\cite{Nakashima:2011fu,ArmendarizPicon:2010rs}.   \n\nAt the qualitative level, one can identify three main effects\ndistinguishing the growth of perturbations in $\\Theta$CDM from that in\n$\\Lambda$CDM. These are: \n{\\it (i)} \n  a rescaling of the matter contribution in the Poisson equation \n(i.e., a different self-gravity of matter perturbations), \n{\\it (ii)} \n  an additional contribution to the anisotropic stress, and \n{\\it (iii)} \n  the presence of additional clustering species.\nThe first two effects are generic for any Lorentz violating\ngravitational theory based on the Einstein-aether or khronometric\nmodel\\footnote{Similar effects are also known in other modified\n  gravity and dark energy models, see e.g.\n  \\cite{Lue:2003ky,D'Amico:2012zv}.},\nwhile the third is specific to the dynamical realization of dark\nenergy in $\\Theta$CDM. \n\n\nTo understand these effects, let us work in the Newtonian gauge. The\nPoisson equation (i.e., the sub-Hubble limit of the $(00)$ Einstein\nequation) reads \n\\begin{equation}\n\\begin{split}\n&k^2(2\\phi-\\alpha\\, \\psi)=-8\\pi G_0 a^2 \\sum_{\\mathrm{other}} \\rho_n \\delta_n\\\\\n&-\\alpha k^2(\\dot\\chi+{\\cal H} (1-B)\\chi)-\\frac{\\alpha\\, G_0}{c^2_\\Theta \\tau_\\a G_{cos}}\n\\left(H_\\alpha \\dot{\\tilde \\xi}-\\frac{\\psi}{\\tau_a}\\right)\\;.\n\\label{eqNewsub}\n\\end{split}\n\\end{equation}\nLet us first concentrate on the contribution of standard matter. Using\nthe Friedmann equation (\\ref{eq:Friedm}), the (time-dependent)\nfraction of the total energy density of the Universe due to each\nmatter component is given by\n\\begin{equation}\n\\label{fract}\nf_n\\equiv\\frac{8\\pi G_{cos}}{3H^2}\\rho_n\\;.\n\\end{equation}\nThe Poisson equation takes the form\n\\begin{equation}\n\\begin{split}\nk^2 \\phi &=\n-4\\pi G_N a^2 \\sum_{\\mathrm{other}} \\rho_n \\delta_n \\\\\n&=\n-\\frac{3}{2}\\frac{G_N}{G_{cos}}{\\cal H}^2 \\sum_{\\rm other}f_n\\delta_n\\;,\n\\label{eqNewsubma}\n\\end{split}\n\\end{equation}\nwhere for the time being we have omitted the terms in the second line\nof (\\ref{eqNewsub}) and neglected the anisotropic stress (i.e. we have\nset $\\psi=\\phi$).  If we assume that the background evolution of the\nUniverse is standard, with \n${\\cal H}$ and $f_n$ \nbeing exactly the same as in $\\Lambda$CDM, Eq.~(\\ref{eqNewsubma})\nimplies that the strength of the gravitational potential produced by\ndensity perturbations is modified by the factor $G_N\/G_{cos}$. When\nthis modified potential is substituted into the matter equations of\nmotion (which have the standard form), it leads to a different growth\nrate of perturbations with respect to the $\\Lambda$CDM case. For\ninstance, a straightforward calculation shows that during the matter\ndominated epoch, the density contrast grows according to the modified\npower-law (cf. \\cite{Kobayashi:2010eh})\n\\begin{equation}\n\\label{deltamat}\n\\delta\\propto a^{\\frac{1}{4}(-1+\\sqrt{1+24 G_N\/G_{cos}})}\\;.\n\\end{equation}\nNotice that for small values of the parameters ($\\a$, $\\b$, $\\l$) the\nanomalous growth is proportional to\n\\begin{equation}\n\\frac{G_N}{G_{cos}}-1= \\frac{\\Sigma}{2}+O(\\alpha^2)\\;,\n\\end{equation}\nwhere we have defined\n\\begin{equation}\n\\label{sigma}\n\\Sigma\\equiv\\a+\\b+3\\l\\;.\n\\end{equation}\nFor $\\Sigma=0$, the effect of modified self-gravity is strongly\nsuppressed. Hence we expect cosmological bounds on ($\\a$, $\\b$, $\\l$)\nto be weaker along this degeneracy direction. Incidentally, in the\nEinstein-aether case $\\Sigma$ is required to vanish (or, rather, be\nextremely small) by the PPN constraints, see (\\ref{ablaether}). This\nexplains why the cosmological bounds on Einstein-aether theory are\nrather mild \\cite{Zuntz:2008zz,Li:2007vz}. On the other hand, in the\nkhronometric model the PPN constraints are compatible with $\\Sigma\n\\neq 0$ and the influence of Lorentz violation on cosmological\nperturbations is more pronounced.\n\nThe second effect is understood from the tracefree part of the $(ij)$\nEinstein equation,\n\\begin{equation}\n\\begin{split}\nk^2&(\\psi-\\phi)=\\\\\n&-12\\pi G_0 a^2\\sum_{\\mathrm{other}}( \\rho_i +p_i)\\sigma_i+\n{\\beta}k^2(\\dot \\chi+2{\\cal H} \\chi).\\label{eqanis}\n\\end{split}\n\\end{equation}\nSince the khronon field introduces a preferred direction $u_\\mu$ in\nspace-time, it may generate anisotropic stress. This effect\n(proportional to $\\beta$) is accounted for by the last term in the\nprevious equation.  Generally speaking, adding anisotropic stress\namounts to increasing the viscosity of the cosmic fluid, and leads to\na damping of small-scale perturbations.\n\nThe third effect comes from gravitational interactions between\nordinary matter species and the scalar fields $\\chi$ and\n$\\tilde{\\xi}$. This interaction, described by the second line in\nEq.~(\\ref{eqNewsub}), may play an important role under the condition\nthat the fields cluster and form sufficiently dense clumps. This might\nbe the case through a mechanism described in Ref.~\\cite{Blas:2011en}.\nThe mixing between the dark energy perturbation $\\tilde\\xi$ and the\nkhronon $\\chi$ (see Eqs.~(\\ref{eqchixi})) gives rise to a mode whose\nsound speed vanishes in the limit of small momentum $k$. This property\nimplies that the effective pressure associated with the mode is small,\nand that the density perturbations ($\\delta \\rho_\\chi$, $\\delta\n\\rho_\\xi$) can be amplified efficiently by gravitational collapse.\nHence, the $\\Theta$CDM model features clustering dark energy. A\nsemi-quantitative analysis of this effect was performed in\n\\cite{Blas:2011en}, showing that structure formation is enhanced at\nsmall comoving momenta (large wavelengths), $k\\lesssim H_\\a$.\nUnfortunately, in this range, the quality of cosmological data is\nrather poor, and this effect does not play a significant role in\nactual observational constraints on the model. \n\nTo illustrate how the above effects impact observable quantities, we\nstudy numerically the evolution of cosmological perturbations in two\nreference models using the Boltzmann code {\\sc class}.  We will call\nthem the \\emph{enhanced gravity} and the \\emph{shear} models. The\ncorresponding parameter values are listed in Table~\\ref{tab:1}.\nClearly, in the \\emph{enhanced gravity} model we keep the effect {\\it\n(i)} while switching off the effect {\\it (ii)}; in the case of the\n\\emph{shear} model the situation is opposite. The effect {\\it (iii)}\nis present in both models but we will see that it is always\nsubdominant. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n Model & $\\a$ & $\\b$ & $\\l$ &$\\Sigma$\\\\\\hline\n\\emph{enhanced gravity} & $0.2$ & $0$ & $0.1$&$0.5$\\\\\n\\hline\n\\emph{shear} & $0.05$ & $ 0.25$ & $-0.1$& 0\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the \\emph{enhanced gravity} and \\emph{shear} models.}\n\\label{tab:1}\n\\end{table}\nNote that the values in Table~\\ref{tab:1} have been chosen very large\nin order to make the modifications visible on the plots.  However,\nthese values are excluded by current data (see Sec.~\\ref{sec:data}).\n\n\n\\subsection{Effects on the CMB}\n\\label{ssec:obserA}\n\nIn this subsection, we describe the changes induced by Lorentz\nviolation in the CMB temperature anisotropy spectrum.  We consider\n$\\Theta$CDM with the two reference sets of parameters listed above and\ncompare the results to $\\Lambda$CDM. To highlight the changes, all\nsimulations are performed with  adjusted initial conditions such that,\nin the limit $k\\rightarrow 0$, the gravitational potential $\\psi$ is\nthe same for all three models. For the standard cosmological\nparameters, we choose the following values:\n$n_s=1$, $h=0.7$, $\\Omega_b=0.05$, $\\Omega_{cdm}=0.25$, $A_s=2.3\\times\n10^{-9}$, $z_{reio}=10$.\\\\\n\n\\noindent\n\\emph{Enhanced gravity model}:\nIn Fig.~\\ref{fig:snapshots} (top panel)\nwe can see that at the time of decoupling, the gravitational potential\nin this model is enhanced (in absolute value) compared to\n$\\Lambda$CDM. \n\\begin{figure}[h!]\n  \\begin{flushleft}\n   \\hspace{-1.3cm}\n    \\includegraphics[scale=0.35]{Figs_pdf\/second_snapshot}\\\\\n    \\vspace{-1.6cm}\n    \\hspace{-1.3cm}\n    \\includegraphics[scale=0.35]{Figs_pdf\/second_snapshot_SW}\n  \\end{flushleft}\n  \\vspace{-1.3cm}\n  \\caption{Top: photon temperature perturbation\n    $\\Theta_\\gamma\\equiv\\delta_\\gamma\/4$ and gravitational potential\n    $-\\psi$ at decoupling, for the three models under scrutiny.\n    Bottom: Sachs-Wolfe contribution, given by $\\Theta_\\gamma+\\psi$\n    (i.e. by the difference between the two curves above).}\n  \\label{fig:snapshots}\n\\end{figure}\nThis is due to an increase in the perturbation growth, governed by\n$\\Sigma$. On the same panel one observes two changes in the solution\nfor the photon temperature perturbations\n$\\Theta_\\gamma\\equiv\\delta_\\gamma\/4$: a shift of the peaks of oscillations\ntowards higher momenta, and a shift of the zero point of oscillations.\nThese effects can be qualitatively understood from the combination of\nthe modified Poisson equation (\\ref{eqNewsubma}) with the equations of\nmotion of the photon-baryon plasma before decoupling. The latter have\na standard form, and in the tight coupling approximation they reduce\nto a single master equation,    \n\\begin{align}\n \\ddot \\Theta_{\\gamma}+\\frac{\\dot R}{1+R}\\dot \\Theta_{\\gamma}+k^2c_s^2\\Theta_{\\gamma} = -\\frac{k^2}{3}\\psi + \\frac{\\dot R}{1+R}\\dot\\phi+ \n \\ddot\\phi,\n  \\label{eqn:ThetaPrimePrime}\n\\end{align}\nwhere $R\\equiv \\frac{3\\rho_b}{4\\rho_\\gamma}$ encodes the\nbaryon-to-photon density ratio, and $c_s\\equiv (3(1+R))^{-1\/2}$ is the\nsound speed of density waves in the plasma in the absence of gravity.\nAccording to Eq.~(\\ref{eqNewsubma}), the first term on the r.h.s. of\n(\\ref{eqn:ThetaPrimePrime}) contains a contribution proportional to\n$\\Theta_\\gamma$. This contribution is positive and larger than in\n$\\Lambda$CDM for $\\Sigma>0$. It effectively decreases\nthe speed of sound in the plasma at the moment when each mode enters\ninside the horizon\\footnote{Strictly speaking, the Poisson equation\n  (\\ref{eqNewsubma}) is valid only for subhorizon modes. However, it\n  is sufficient for our qualitative argument.},  \nwhich translates into a shift of the peaks, observed in the top panel\nof Fig.~\\ref{fig:snapshots}.  Next, the zero point of the acoustic\noscillations is given by the value of $-\\psi\/3c_s^2$ at decoupling. As\nalready mentioned, $|\\psi|$ is larger  in the {\\it enhanced gravity}\nmodel, leading to a shift in this zero point further away from\n$\\Theta_\\gamma=0$. Finally, the amplitude of the acoustic\noscillations around the zero point is slightly smaller in the {\\it\nenhanced gravity} model than in $\\Lambda$CDM.\n\nThe above features affect the Sachs--Wolfe (SW) contribution to\ntemperature anisotropies. In the bottom panel of\nFig.~\\ref{fig:snapshots}, we plot this contribution, given by\n$\\Theta_{\\gamma}+\\psi$ at recombination.  One clearly sees the shift\nof the peaks and notices that the even peaks are suppressed while the\nodd ones are almost constant (with the notable exception of the first\npeak, which is suppressed). This is due to the competition between\nthe different effects described in the previous paragraph.\n\nIn the anisotropies observed today, the SW contribution is\nsupplemented by those coming from the Doppler and Integrated\nSachs--Wolfe (ISW) effects. The decomposition of the $C_\\ell$ spectrum\nin terms of these effects is presented in\nFig.~\\ref{fig:cl_decomposition} for different models.  The comparison\nof the \\emph{enhanced gravity} model with $\\Lambda$CDM is shown in the\ntop-panel.  In the SW contribution (as well as in the total spectrum),\nwe observe the expected suppression of the first, second and fourth\npeaks, while the third peak amplitude is roughly unchanged.  On small\nangular scales, the peaks are further suppressed by Silk damping.\nIndeed, due to the shift in the phase of oscillations, they correspond\nto smaller physical scales at recombination, that are more affected by\ndiffusion damping.  The Doppler effect, which depends on\n$\\dot\\Theta_\\gamma$ at recombination, is also modified. Finally, a\nprominent feature clearly visible on the plot is the significant\nenhancement of the ISW contribution in the range $10<l<100$, i.e.\nbetween the regions usually affected by the early ISW effect\n($100<l<200$) and the late ISW effect ($2<l<10$).  The ISW effect is\nproportional to the time derivative of the gravitational potential. In\nthe $\\Lambda$CDM model, the potential varies only during the epochs of\nradiation and $\\Lambda$ domination. However, in the \\emph{enhanced\ngravity} model, the growth of density perturbations entails a slow\nincrease of the gravitational potential also during the matter\ndominated era, enhancing the ISW effect on a wide range of scales.\n\nAll in all, we conclude that the \\emph{enhanced gravity} model\nproduces significant modifications in the spectrum of CMB\nanisotropies. The pattern of these modifications is quite specific,\nand apparently not degenerate with the effects of standard\ncosmological parameters. \\\\\n\\begin{figure}[h]\n  \\begin{flushleft}\n    \\hspace{-1.3cm}\n    \\includegraphics[scale=0.35]{Figs_pdf\/cl_decomposition_nobeta}\\\\\n    \\vspace{-1.6cm}\n    \\hspace{-1.3cm}\n    \\includegraphics[scale=0.35]{Figs_pdf\/cl_decomposition_zerosum}\n  \\end{flushleft}\n  \\vspace{-1.3cm}\n  \\caption{\n     Temperature anisotropy spectrum (solid) and its decomposition in\n     terms of Sachs--Wolfe (dotted), Doppler (dashed) and Integrated\n     Sachs--Wolfe (dot-dashed) contributions. For clarity, we do not\n     show cross correlations between these contributions.  Thick black\n     lines represent the $\\Lambda$CDM model, while thin blue lines are\n     used for the two $\\Theta$CDM reference models.}\n  \\label{fig:cl_decomposition}\n\\end{figure}\n\n\\noindent\n\\emph{Shear model}:\nWe recall that in this model, the $\\chi$ field generates some\nanisotropic stress and contributes to the shear of the perturbed\nmetric, as described by Eq.~(\\ref{eqanis}).  The presence of shear\ntends to smooth out metric perturbations on scales smaller than the\nsound horizon associated with the sound speed of the $\\chi$ field.\nNote that for parameters of the particular \\emph{shear} model studied\nhere, the field $\\chi$ is superluminal\\footnote{As pointed above, this\n  does not present any inconsistencies in theories without Lorentz\n  invariance.}\n($c_\\chi=\\sqrt{3}$) and the suppression appears already on\nsuper-Hubble scales.\n\nThis is indeed observed on the top panel of Fig.~\\ref{fig:snapshots},\nwhere the gravitational potential is clearly smaller (in absolute\nvalue) compared to $\\Lambda$CDM. It also exhibits notable wiggles\ncaused by oscillations in the $\\chi$-field \\cite{Blas:2011en}.  The\nfact that $c_\\chi$ is much larger than the photon-baryon sound speed\nexplains the shift between the phase of the oscillations seen in\n$\\psi$ and in $\\Theta_\\gamma$.  The suppression of $\\psi$ shifts the\nzero-point of the oscillations in the photon temperature $\\Theta_\\gamma$.\nOn the other hand, we do not see any shift in the positions of the\npeaks, which is compatible with the previous discussion: the\nself-gravity of radiation is not modified in this model.  \n \nFor fixed initial conditions $\\Theta_\\gamma(\\tau_0)$, the amplitude of\nthe acoustic oscillations depends crucially on boosting effects,\nimprinted around the time of Hubble crossing, and caused by the three\ngravitational driving terms on the right-hand side of\nEq.~(\\ref{eqn:ThetaPrimePrime}). In the \\emph{shear} model, the\namplitude of acoustic oscillations is damped as a consequence of\nsmaller metric fluctuations and reduced gravitational boosting.  \nThis translates into an overall suppression\nof the SW effect visible in the bottom panel of\nFig.~\\ref{fig:snapshots}.\n\nOn Fig.~\\ref{fig:cl_decomposition}, we see that the Doppler and ISW\ncontributions to the total temperature spectrum $C_\\ell$ are also\nlower in the \\emph{shear} model than in $\\Lambda$CDM. The net result\nis a uniform suppression of all peaks.  One may expect that this\neffect could be compensated, at least partially, by a rescaling of the\ninitial amplitude of perturbations. This suggests that pure\n\\emph{shear} models might be less constrained than \\emph{enhanced\ngravity} ones.\n\nOur \\emph{shear} model is similar to the one studied in\n\\cite{Zuntz:2008zz}, where it was claimed that the dominant effect on\nthe CMB comes through the ISW. Our analysis demonstrates that the\nchanges in the SW and Doppler contributions are equally important for\nthis model.\n\n\n\\subsection{Effects on the matter power spectrum}\n\nAnother observable affected by the behaviour of cosmological\nperturbations in $\\Theta$CDM is the matter power spectrum. We restrict\nthe discussion to low enough Fourier modes, $k\\lesssim 0.1~$h\/Mpc, for\nwhich perturbations are still very close to the linear regime. The\nstudy of non-linear clustering in this model is beyond the scope of\nthe present paper. \n\nThe definition of what one calls the ``matter power spectrum'' is not\nso obvious in $\\Theta$CDM.  This model contains additional components that\ncontribute to perturbations of the total energy density.  One may\nwonder whether they must be included in the calculation of the power\nspectrum. In general, the answer to this question is yes. \nIndeed, the existing observations fall into two categories. The first\ncategory probes directly metric perturbations (themselves related to\ntotal density fluctuations): this is the case e.g. for cosmic shear\nsurveys or CMB lensing measurements. The second category measures the\nclustering of compact objects like galaxies, halos or clusters. On\nlarge scales, these objects are known to trace linearly the underlying\ngravitational field~\\cite{Bernardeau:2001qr}.  Thus, it appears\nreasonable to \\emph{define} the matter power spectrum using the\nPoisson equation\\footnote{Strictly speaking, this definition makes\n  sense only for modes well inside the horizon, where the Poisson\n  equation is valid.  However, this is sufficient for our study, because\n  the power spectrum is actually measured for such modes only.}  \n\\begin{equation}\n\\label{drhodef}\nk^2\\phi=-4\\pi G_Na^2\\delta\\rho_{tot}\\;.\n\\end{equation}\nNote that we have used here the locally determined value of the Newton\nconstant introduced in Eq.~(\\ref{gn}). Comparing Eq.~(\\ref{drhodef})\nto Eq.~(\\ref{eqNewsub}) and neglecting the anisotropic stress which is\nvery small on relevant scales, we see that the contribution of\nordinary matter to $\\delta\\rho_{tot}$ coincides with the standard\ndefinition $\\delta\\rho_n$, while the contributions of the khronon and\n$\\tilde\\xi$ fields read\n\\begin{equation}\\label{newtpois}\n\\begin{split}\n&\\delta\\rho_\\chi\\equiv \\frac{\\alpha\\, k^2}{8\\pi a^2G_0}  (\\dot \\chi+\\mathcal H (1-B)\\chi)\\;,\\\\\n&\\delta\\rho_\\xi\\equiv  \\frac{\\a}{8\\pi a^2\n  G_{cos}c_\\Theta^2\\tau_\\alpha}\n\\bigg(H_\\a\\dot{\\tilde \\xi}-\\frac{\\psi}{\\tau_\\a}\\bigg)\\;.\n\\end{split}\n\\end{equation}  \nWe stress that all quantities here are taken in the Newtonian gauge. \n\n\\begin{figure*}[!t]\n  \\begin{center}\n    \\includegraphics[scale=0.3]{Figs_pdf\/lowk_delta_rhos_nobeta_newt}\\hspace{-1cm}\n    \\includegraphics[scale=0.3]{Figs_pdf\/lowk_delta_rhos_zerosum_newt}\\vspace{-1cm}\\\\\n    \\includegraphics[scale=0.3]{Figs_pdf\/midk_delta_rhos_nobeta_newt}\\hspace{-1cm}\n    \\includegraphics[scale=0.3]{Figs_pdf\/midk_delta_rhos_zerosum_newt}\\vspace{-1cm}\\\\\n    \\includegraphics[scale=0.3]{Figs_pdf\/highk_delta_rhos_nobeta_newt}\\hspace{-1cm}\n    \\includegraphics[scale=0.3]{Figs_pdf\/highk_delta_rhos_zerosum_newt}\\vspace{-1cm}\n  \\end{center}\n  \\caption{ Evolution of the transfer function of various energy\n    density perturbations $\\delta \\rho_i$ (arbitrary units) in the\n    Newton gauge, for the \\emph{enhanced gravity} model (left) and the\n    \\emph{shear} model (right), and for the three wavenumbers\n    $k=10^{-7}$Mpc$^{-1}$ (top panels), $k=6\\times 10^{-4}$Mpc$^{-1}$\n    (middle panels) and $k=0.1$Mpc$^{-1}$ (bottom panels). For\n    comparison, we show the evolution of $\\delta \\rho_b$ and $\\delta\n    \\rho_{cdm}$ in the $\\Lambda$CDM case. We also indicate the\n    conformal time of Hubble crossing for the two largest \nwavenumbers,\n    and the conformal time of radiation to matter equality.}\n  \\label{fig:delta_rhos}\n\\end{figure*}\n\nTo obtain the matter power spectrum, the density perturbation\n$\\delta\\rho_{tot}$ must be divided by the total matter density (dark\nmatter plus baryons). The latter is determined from the background\ncosmological solution. In the case of standard gravity, one would\nagain use the local Newton constant $G_N$ to infer the density from\nthe geometry.  Because in the $\\Theta$CDM model the gravitational constant\nin the Friedmann equation is different, the effective matter density\nfound in this way is renormalized compared to the actual value,\n\\begin{equation}\n\\rho_\\mathrm{eff}=\\frac{G_{cos}}{G_N}(\\rho_{cdm}+\\rho_b)\\;.\n\\end{equation}\nThus we arrive at the following formula for the power spectrum,\n\\begin{equation}\n\\label{Pkdef}\nP(k)=\n\\bigg(\\frac{G_N}{G_{cos}}\\bigg)^2\n\\frac{\\langle | \\delta\\rho_{tot}(\\vec{k}) |^2 \\rangle}{(\\rho_{cdm}+\\rho_b)^2}\\;.\n\\end{equation}\n Let us discuss the imprint of our two reference $\\Theta$CDM models on this spectrum.\n\\\\\n\n\\noindent\n\\emph{Enhanced gravity model}:\nThe left panels of Fig.~\\ref{fig:delta_rhos} show the time dependence\nof various contributions to $\\delta\\rho_{tot}$ in this model, for\nthree values of the momentum $k$. As expected, inside the Hubble\nradius, matter density perturbations grow at a larger rate and are\nenhanced compared to the $\\Lambda$CDM case.  The contribution of the\n$\\chi$ and $\\xi$ fields, though small, exhibits some interesting\nfeatures.  On super-Hubble scales, the $\\xi$-density perturbation\nrapidly grows with time.  This is due to the mixing between $\\chi$ and\n$\\tilde{\\xi}$ discussed in Ref.~\\cite{Blas:2011en}, which gives rise\nto a mode with vanishing sound speed at low momenta, and allows for\nclustering of the dark energy.  In the case of modes crossing the\nHubble scale around the current epoch, $\\delta\\rho_\\xi$ even becomes\ncomparable to the matter contributions at the present time.  However,\nat shorter scales, the mixing between $\\chi$ and $\\tilde{\\xi}$\ndisappears and the speeds of sound of these components become non\nnegligible\\footnote{We remind that in this work we consider the regime\n  where $c_\\chi$, $c_\\Theta$ are of order one. If instead these\n  velocities were very small, the fields $\\chi$ and $\\tilde{\\xi}$\ncould in principle cluster also on scales much smaller than the Hubble\nradius.}. This leads to damped oscillations of these fields, clearly\nvisible in the lower left panel of Fig.~\\ref{fig:delta_rhos}.\n\\begin{figure}[h]\n  \\begin{flushleft}\n    \\hspace{-1.25cm}\n    \\includegraphics[scale=0.35]{Figs_pdf\/pk_ratio}\n    \\vspace{-1.5cm}\n  \\end{flushleft}\n  \\caption{Ratios of the matter power spectra in the two reference\n    $\\Theta$CDM models and in $\\Lambda$CDM at redshift $z=0$.}\n  \\label{fig:pk_ratio}\n\\end{figure}\n\nThe ratio of the power spectra in the \\emph{enhanced gravity} model\nand in $\\Lambda$CDM is presented in Fig.~\\ref{fig:pk_ratio}. The\naccelerated growth of matter density perturbations translates into a\nsignificant scale-dependent enhancement of the power spectrum on\nscales that are below the Hubble radius today (i.e. with $k\\gtrsim\n0.0003\\,h$\/Mpc).  The curve exhibits small wiggles due to a shift in\nthe position of the peaks of baryon acoustic oscillations (cf. the\nshift of CMB peaks discussed in Sec.~\\ref{ssec:obserA}).  On\nsub-Hubble scales, the contribution of $\\delta\\rho_\\chi$ and\n$\\delta\\rho_\\xi$ to the total matter power spectrum is\nnegligible\\footnote{As mentioned above, this conclusion could change\nin a scenario where a tiny value of $c_\\chi$ or $c_\\Theta$  would be\nassumed.}.  Finally, let us point out that the curve in\nFig.~\\ref{fig:pk_ratio} must be taken with a grain of salt at\n$k\\gtrsim 0.1\\,h$\/Mpc, where non-linearities become important.\\\\ \n\n\\noindent\n\\emph{Shear model}: \nOur definition of the matter power spectrum refers to sub-Hubble\nwavelengths only, and also uses the assumption that $\\phi$ and $\\psi$\nare equal in Eq.~(\\ref{eqNewsub}), which leads to Eq.~(\\ref{drhodef}).\nHence we need to check whether the anisotropic stress (responsible for\na possible difference between the two metric perturbations, see\nEq.~(\\ref{eqanis})) vanishes inside the Hubble radius in the shear\nmodel. It is well-known that the anisotropic stress of neutrinos and\nof decoupled photons decays inside the Hubble radius, due to\nfree-streaming. This conclusion also applies to the anisotropic stress\nof the khronon, due to the dynamics of the $\\chi$ field. For instance,\nduring matter domination, $\\chi$ oscillates with an envelope decaying\nlike $\\tau^{-3}$ on sub-Hubble scale, implying that its averaged\nenergy density decays like $\\delta \\rho_\\chi \\propto \\tau^{-8} \\propto\na^{-4}$. The contribution of the khronon anisotropic stress to the\ndifference $(\\phi-\\psi)$ is then oscillating with an envelope\nproportional to $\\tau^{-4} \\propto a^{-2}$. At the same time, $\\phi$\nand $\\psi$ are almost constant. Hence the relative difference between\n$\\phi$ and $\\psi$ quickly becomes negligible, and our definition of\nthe matter power spectrum on sub-Hubble scale is applicable.\n\nIf we normalize perturbations to the same initial\nvalue of $\\psi$, the evolution of ($\\delta\\rho_{cdm}$, $\\delta\\rho_b$)\nis identical above the Hubble scale in the $\\Lambda$CDM and\n\\emph{shear} models. However, in the \\emph{shear} case, the\nanisotropic stress of the khronon cannot be neglected soon after the\ntime of Hubble crossing.  This leads to a smoothing of metric\nperturbations and to a suppression of $\\delta\\rho_{cdm}$ and\n$\\delta\\rho_b$ that is clearly visible in the lower right panel of\nFig.~\\ref{fig:delta_rhos}. Well after Hubble crossing, the khronon\nanisotropic stress becomes negligible and ($\\delta\\rho_{cdm}$,\n$\\delta\\rho_b$) evolve like in $\\Lambda$CDM, but with a constant\noffset coming from the suppression experienced soon after Hubble\ncrossing.\n\nThis suppression is seen better in Fig.~\\ref{fig:pk_ratio}. For\nthe chosen parameter values, the effect of the \\emph{shear} model  is\nsmaller than for the \\emph{enhanced gravity} model. In order to see\nclearly both effects with the same scale, we multiplied the difference\nbetween the power spectra in the \\emph{shear} model and in\n$\\Lambda$CDM by 20. Inside the Hubble radius, the suppression of the\npower spectrum in the \\emph{shear} model is almost scale-independent.\nWhen approaching the current value of the Hubble scale, the\nsuppression is reduced by the counteracting effect of additional\ndensity fluctuations in the $\\xi$ field. However, this reduction is\nonly significant on scales that are too large to be observed with\ngood precision (due to the sampling variance associated to a given\nsurvey). Besides, on such scales our definition of the matter power\nspectrum is no longer \napplicable\\footnote{Like in the   \\emph{enhanced\ngravity} model, we note that in a scenario with tiny values of\n$c_\\Theta$ or $c_\\chi$, density fluctuations in the $\\tilde{\\xi}$ or\n$\\chi$ field could in principle be significant on much smaller\nscales.}.\n\nWe reach the same conclusion as in the CMB case: on observable scales,\nthe effect of pure \\emph{shear} models can be mimicked by an overall\nreduction of the primordial fluctuation amplitude, which suggests that\nthese models are more weakly constrained than \\emph{enhanced gravity}\nmodels.\n\n\\section{Comparison with current data}\\label{sec:data}\n\nWe will now compare the khronometric $\\Theta$CDM model to CMB  and LSS\ndata, using the parameter inference code {\\sc Monte\nPython}\\footnote{\\tt http:\/\/montepython.net} \\cite{Audren:2012wb}.\nFor the CMB, we use here WMAP 7-year data \\cite{Komatsu:2010fb}, and\nSPT data from 2008 and 2009 \\cite{Keisler:2011aw}.  Updating our\nanalysis with recent Planck data \\cite{Ade:2013ktc} could give a small\nimprovement on parameter constraints, without changing their order of\nmagnitude. For LSS, we rely on galaxy power spectrum data from the\nWiggleZ redshift survey \\cite{Parkinson:2012vd}. We choose to perform\nour runs in the synchronous gauge. \n\nWe include in the fit eight free cosmological parameters, which\nare the usual six free parameters  of the minimal flat $\\Lambda$CDM\nmodel, plus $\\beta$ and $\\beta+\\lambda$ (this combination is chosen to\nfacilitate the convergence of the chains).  The parameter $\\alpha$ is\nfixed to $2\\beta$, which is the condition for satisfying all bounds\ncoming from Solar System tests in the khronometric model\\footnote{We\n  decided to focus only on the khronometric case, since from the\n  comments after Eq.~\\eqref{sigma} it is clear that the bounds are\n  stronger in this model than in the Einstein-aether case. This is\n  confirmed by comparing our results with the constraints presented in\n  Ref.~\\cite{Zuntz:2008zz}.}.\nAs far as cosmology is concerned, this condition is actually not\nnecessary, but we implement it anyway in order to deal only with\nrealistic models.  We also vary three nuisance parameters describing\nthe foreground contamination of SPT data, and we marginalize over\nthese parameters following strictly the approach of\nRef.~\\cite{Keisler:2011aw}.\n\nWe show our results for the Bayesian minimum credible interval of\neach parameter in Table~\\ref{table}. The one-dimensional and\ntwo-dimensional posterior parameter distributions are displayed in\nFig.~\\ref{fig:triangle} (omitting the three SPT nuisance parameters\nfor clarity).\n\n\\begin{table*}\n  \\begin{tabular}{|cccccccc|}\n     \\hline$100~\\omega_{b }$  & $\\omega_{cdm }$  & $n_{s }$  &\n     $10^{+9}A_{s }$  & $h$  & $z_{reio }$  & $\\beta$  &\n     $\\beta+\\lambda$  \\\\ \\hline \n     \\vspace{-0.1cm}\n     &&&&&&&\\\\\n     $2.219_{-0.044}^{+0.041}$ & $0.1205_{-0.0029}^{+0.0027}$ &\n     $0.9552_{-0.011}^{+0.011}$ & $2.556_{-0.092}^{+0.09}$ &\n     $0.6782_{-0.013}^{+0.013}$ & $10.05_{-1.2}^{+1.2}$ &\n     $<0.05$ & $<0.012$ \\\\[0.2cm]\n     \\hline\n   \\end{tabular}\n\\caption{Mean values and 68\\% confidence limits of the minimum credible\ninterval of the parameters of the khronometric $\\Theta$CDM model. For $\\beta$ and\n$\\beta+\\lambda$, we show instead 95\\% upper limits.}\n\\label{table}\n\\end{table*}\n\n\\begin{figure*}\n  \\includegraphics[scale=0.45]{Figs_pdf\/lcdm_nde_alpha_2_beta_sum_triangle.pdf}\n  \\caption{Triangle plot showing the one-dimensional marginalized\n    posterior distribution and the two-dimensional probability contours\n    (at the 68, 95 and 99\\% confidence level) of the parameters of the\n    khronometric $\\Theta$CDM model.}\n\\label{fig:triangle}\n\\end{figure*}\n\nThe $\\Theta$CDM parameters  are found to be weakly correlated with the\nsix standard model parameters, and strongly correlated with  each\nother. Isoprobability contours in the $(\\beta, \\beta+\\lambda)$ space\nhave a complicated shape. They are very elongated along two directions\nof degeneracy, corresponding to $\\beta+\\lambda \\simeq 0$ and $\\beta\n\\simeq 3\/2(\\beta+\\lambda)$. Away from these two special directions,\nthe contours are more regular, and closer to an ellipse centered on\nthe origin. The posterior probability peaks at $(\\beta,\n\\beta+\\lambda)\\simeq (0,0)$, showing that the data brings no evidence\nin favor of the $\\Theta$CDM model.  The two directions of degeneracy\ncan be interpreted as follows. \n\nFirst, the case $\\beta+\\lambda \\simeq 0$ corresponds to  $\\Sigma\n\\simeq 0$ for our choice of $\\alpha$. This is the region of pure\n\\emph{shear} models, discussed in the previous subsection. We already\nargued that the differences with $\\Lambda$CDM are less important in\nthis case\\footnote{Under the condition $\\alpha=2\\beta$, the velocity of the\n  $\\chi$-field is given by $c^2_\\chi=\\Sigma\/(6\\beta)$. Thus, the limit\n  $\\Sigma=0$ formally corresponds to vanishing sound speed of the\n  $\\chi$-component. One could expect that this would lead to an amplification of\n  the $\\chi$-perturbations and, as a consequence, to strong constraints on the\n  model. However, in practice, $c_\\chi$ never becomes smaller than $\\simeq\n  10^{-2}$ during the Monte Carlo sampling of the parameters and the clustering\n  of $\\chi$ does not appear to be significant.}\nthan in the \\emph{enhanced gravity} case corresponding to\nthe orthogonal direction $\\beta=0$. This explains the factor four\ndifference between constraints on $\\beta$ and on $\\beta+\\lambda$ in\nTable~\\ref{table}.\n\nThe other case has a less straightforward interpretation. The relation\n$\\beta \\simeq 3\/2(\\beta+\\lambda)$ implies that the khronon has a\nsquared sound speed $c_\\chi^2 \\simeq 1\/3$, similar to that of\nneutrinos and of tightly coupled photons. Hence, for these models, the\nkhronon does not introduce a new characteristic scale, and the \noscillations seen in the lower left panel of Fig.~\\ref{fig:delta_rhos} may \n mimic those of photons or neutrinos, thus lowering the\nsensitivity along this direction. We checked explicitly with the\nmodified {\\sc class} code that in this particular case, the sum of all\neffects described in the previous section (enhanced gravity,\nadditional shear and extra clustering species) can be nearly cancelled\nat the level of CMB anisotropies by a variation of the standard\ncosmological parameters. This situation is analogous to the case of\nextra relativistic degrees of freedom, for which the impact of a small\nvariation of $N_\\mathrm{eff}$  on the CMB can be partially compensated\nby a variation of other parameters, leading to a well-known degeneracy\nbetween $N_\\mathrm{eff}$ and $H_0$~\\cite{book}. $\\Theta$CDM models\nwith $c_\\chi^2 \\simeq 1\/3$ are not exactly equivalent to models with\nextra relativistic degrees of freedom, but their effect can be\ncompensated in a similar way. This explains the weak correlation\nobserved in Fig.~\\ref{fig:triangle} between standard parameters and\n$\\Theta$CDM parameters. By comparing with a run based on CMB data\nonly, we find that the inclusion of LSS data helps in breaking this\ndegeneracy (by limiting enhanced gravity effects on the matter power\nspectrum), but even in presence of WiggleZ data, the degeneracy\nappears very clearly in Fig.~\\ref{fig:triangle}.\n\n\n\\section{Conclusions}\\label{sec:concl}\n\nIn this work, we described the impact of the $\\Theta$CDM model  on\ncosmological observables, on scales where linear cosmological\nperturbation theory is valid.  This model is an alternative to the\n$\\Lambda$CDM scenario in which the acceleration of the universe\nexpansion is caused by a dynamical scalar field $\\Theta$.  An\nimportant difference between this model and most quintessence models\nis that the action of the field $\\Theta$ is naturally protected from\nlarge ultraviolet corrections.  In addition, the $\\Theta$CDM model is\nembedded in a family of gravitation theories with broken Lorentz\ninvariance, which may have an ultraviolet completion in the framework\nof Ho\\v rava gravity.  The latter theories are characterized by a\ntime-like dynamical vector field $u^\\m$ which defines a preferred time\ndirection. Typical examples are the Einstein-aether and khronometric\ntheories. The  $\\Theta$CDM model is the simplest extension of these\ntheories to include a mechanism for cosmic acceleration.  Under simple\nconstraints on its parameters, it passes all local tests of gravity:\nSolar System dynamics, gravity wave emission, black hole structure.\nWe have shown that the $\\Theta$CDM model may produce observable\neffects on cosmological scales, constrained by current cosmological\ndata, but still potentially detectable with future ones.\n\nThe background evolution in $\\Theta$CDM is identical to $\\Lambda$CDM.  The\nonly potential difference could come from a component playing the role\nof stiff matter, but in order to preserve primordial nucleosynthesis,\nwe must assume that this component is too small for playing any role\nin late-time cosmology, and for affecting CMB and large scale\nstructure observables. However, \nthe  evolution of cosmological perturbations is\n generically\nvery different in the $\\Theta$CDM and $\\Lambda$CDM models.  We studied the\nevolution equations of scalar perturbations in $\\Theta$CDM and identified\nthree different effects: \n{\\it (i)} \na renormalization of the matter contribution to the Poisson equation \n(i.e. a different self-gravity of matter perturbations), \n{\\it (ii)} \na new contribution to the anisotropic stress, and \n{\\it (iii)} \nthe presence of additional clustering degrees of freedom. The first\ntwo effects are generic for Lorentz violating gravitation theories\nbased on a unit time-like vector, while the third one is specific to\nthe dynamical realization of dark energy in $\\Theta$CDM. \n\nWe implemented the equations of $\\Theta$CDM in the Boltzmann code  {\\sc\nclass} \\cite{Blas:2011rf}, in order to compute accurately the impact\nof the three effects {\\it (i)} , {\\it (ii)} , {\\it (iii)}. We found\nthat they affect the power spectrum of Cosmic Microwave Background\nanisotropies in a very particular way (shift in the position and\namplitude of the peaks, and enhanced ISW), as can be seen in\nFig.~\\ref{fig:cl_decomposition}. Furthermore, they influence the\nshape of the matter power spectrum (different amplitude and slope on\nobservable scales, shift in the position of baryon acoustic\noscillations).  These effects are shown in\nFigures~\\ref{fig:delta_rhos} and \\ref{fig:pk_ratio}.\n\nTo derive constraints on the free parameters of the khronometric\n$\\Theta$CDM model (considering only parameter combinations satisfying\nSolar System tests), we used data from WMAP (7-year), from SPT (2008\nand 2009 observations), and from the WiggleZ redshift survey.  We ran\nthe parameter inference code {\\sc Monte Python} \\cite{Audren:2012wb}\nand found the bounds displayed in Table~\\ref{table} and illustrated in\nFig.~\\ref{fig:triangle}.  Quite remarkably, these bounds constrain\ndeviations from general relativity to better than the percent level.\nThey are stronger than those from several tests of gravitation, such\nas radiation damping of binary systems, or BBN.  They are also\nstronger for the khronometric case studied in this paper than for the\nEinstein-aether model considered in Ref.~\\cite{Zuntz:2008zz}, due to\nthe absence of the effect {\\it (i)} in the latter case.  The vanishing\nof {\\it (i)} in the Einstein-aether theory (once Solar System bounds\nare imposed) seems to be simply a coincidence.\n\nThe fact that our bounds are dominated by the effect {\\it (i)} -- modified\nself-gravity of the matter perturbations -- suggests that they may have a wider\napplication. The bound on the combination $\\beta+\\lambda$ in Table II can be\ncast into the constraint on the discrepancy between the gravitational constants\nappearing in the Newton law and in the Friedmann equation, $|G_N\/G_{cos}-1| <\n0.018$ at $95 \\%$ confidence level.  Though in our model we cannot completely\ndisentangle {\\it (i)} from other effects, we believe that the above constraint\nwill apply, at least by order of magnitude, to any theory predicting a\ntime-independent discrepancy between $G_N$ and $G_{cos}$. It\nrepresents an improvement compared to the bounds existing in the literature\n\\cite{Robbers:2007ca,Bean:2010zq}.  \n\nOur analysis can be extended in several ways.  First, we have only\nused the scalar sector of the theory. For the Einstein-aether case,\nthe theory contains also propagating vector modes, whose influence on\nB-type CMB polarization could provide further observational\ntests~\\cite{ArmendarizPicon:2010rs,Nakashima:2011fu}.  Second, the\nanalysis can be updated with CMB data from Planck, and in the future,\nwith large scale structure data from DES, LSST or Euclid.  It would\nalso be extremely interesting to go beyond the linear regime and try\nto understand the consequences of $\\Theta$CDM for non-linear structure\nformation.  Finally, to completely characterise possible deviations\nfrom Lorentz invariance in the context of cosmology, one can consider\nthe option that this symmetry is violated also in the dark matter\nsector~\\cite{Blas:2012vn}.\n\n\\section*{Acknowlegements}\nWe are grateful to Mikhail Ivanov, Gregory Gabadadze, Roman\nScoccimarro and Takahiro Tanaka for useful discussions. D.B. and S.S.\nthank the organizers and participants of the Kavli IPMU Focus Week on\nGravity and Lorentz Violations for the encouraging interest and\nvaluable comments.  S.S. thanks the Center for Cosmology and Particle\nPhysics of NYU for hospitality during the completion of this work.\nThis work was supported in part by the Grant of the President of\nRussian Federation NS-5590.2012.2 (S.S.), the Russian Ministry of\nScience and Education under the contract 8412 (S.S.), the RFBR grants\n11-02-01528 (S.S.), 12-02-01203 (S.S.) and by the Dynasty Foundation\n(S.S.). J.L. and B.A. acknowledge support from the Swiss National Science\nFoundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe say that a natural number $n$ is $y$-smooth if all of its prime factors are $\\le y$. \nLet $\\Psi(x,y)$ be the number of such $n\\le x$. \nImproving on many earlier efforts by a number of different authors,\nMatom\\\"aki and Radziwi\\l\\l \\cite{MR} established the existence of many $x^\\epsilon$-smooth numbers \nin intervals of the form $[x,x+c(\\epsilon) \\sqrt{x}]$, for every $\\epsilon>0$. \nHarman \\cite{Har} showed that intervals around $x$ of length $x^{0.45...}$ contain many $x^{0.27...}$-smooth numbers.  \n\nWe are interested in the existence of $x^a$-smooth numbers in much shorter intervals, when\n$a> 1\/2$. More precisely, given $a \\in (1\/2, 1)$, how small can we take $b$ such that  \n\\begin{equation*}\\label{qeq}\n\\Psi(x,x^a)-\\Psi(x-x^b,x^a) \\gg  x^{b-\\epsilon}\n\\end{equation*}\nfor every $\\epsilon>0$?\nIn that direction, Friedlander and Lagarias \\cite{FL} showed that there exists a constant $c>0$ such that \n$b=1-a - c a(1-a)^3$ is admissible, even with $\\epsilon=0$, but without providing any numerical estimate for $c$. \nWe will use exponent pairs (see \\cite{GK}) to find explicit values of $b<1-a$. \nIn particular, $b=1-a-a(1-a)^3$ is admissible for every $a \\in  (1\/2,1)$.\n\nLet $\\psi(x)=x-\\lfloor x\\rfloor -1\/2 = \\{x\\}-1\/2$. \nThe method used by Friedlander and Lagarias \\cite{FL} starts with Chebyshev's identity and \nrequires estimates for sums of $\\psi(x\/p) \\log p$, where $p$ runs over primes. \nOur approach involves sums of $\\psi(x\/n)$ over all integers $n$ from an interval.\nWe use the estimate\n\\begin{equation}\\label{UB}\n\\sum_{N\\le n \\le 2N} \\psi(x\/n) \\ll \\min\\bigl(x^\\theta , x^{k\/(k+1)} N^{(l-k)\/(k+1)} \\bigr)  \\qquad (1\\le N \\le \\sqrt{x}),\n\\end{equation}\nwhere $(k,l)$ is any exponent pair. The two most recent records for $\\theta$ are  $\\theta=\\frac{131}{416}+\\epsilon =0.3149...$ by Huxley \\cite[Thm. 4]{Hux} and $\\theta=\\frac{517}{1648}+\\epsilon=0.3137...$ by Bourgain and Watt \\cite[Eq. (7.4)]{BW}.\nFor the second estimate in \\eqref{UB}, see Graham and Kolesnik \\cite[Lemma 4.3]{GK}. \n\nLet $\\nu=2.9882...$ be the minimum value of $(2^u-1)\/(u-1)$ for $u>1$. \n\\begin{theorem}\\label{thm1}\nLet $(k,l)$ be an exponent pair and $\\theta$ as in \\eqref{UB}.\nThere is a constant $K$ such that \n$$\\Psi(x,y)-\\Psi(x-z,y) \\gg \\frac{z}{(\\log x)^\\nu},$$ \nprovided $x \\ge y \\ge \\sqrt{2x}$ and $x\\ge z\\ge K \\min\\bigl(x^\\theta,x^{l\/(k+1)}y^{(k-l)\/(k+1)}\\bigr)$.\n\\end{theorem}\n\nDefine\n\\begin{equation}\\label{bdef}\nb=b(a,k,l) = \\frac{l+a(k-l)}{k+1}.\n\\end{equation}\n\n\\begin{corollary}\\label{cor1}\nLet $(k,l)$ be an exponent pair, $\\theta$ as in \\eqref{UB} and $1\/2< a\\le   1$.\nThere is a constant $K$ such that for $x\\ge z\\ge K x^{\\min(\\theta ,b)}$,\n$$\\Psi(x,x^a)-\\Psi(x-z,x^a) \\gg \\frac{z}{(\\log x)^\\nu}.$$ \nIf $a=1\/2$, the conclusion holds if $x^a$ is \nreplaced by $\\sqrt{2x}$.\n\\end{corollary}\n\nStarting with the exponent pair $(\\kappa,\\lambda)=(13\/84+\\epsilon,55\/84+\\epsilon)$ \nof Bourgain \\cite[Thm. 6]{Bourgain}, and possibly applying van der Corput's processes $A$ or $B$,\nwe find a sequence of linear functions in $a$, shown in Table \\ref{table1}.\nWhen $a$ is close to $1\/2$, then $\\theta$ is smaller than any $b$ obtained from known exponent pairs. \nWhen $a$ is close to $1$, we rely on exponent pairs $(k,l)$ with small $k$. \nHeath-Brown \\cite[Thm. 2]{HB} found that for integers $m\\ge 3$ and every $\\epsilon >0$,\n\\begin{equation}\\label{hbeq}\nk_m=\\frac{2}{(m-1)^2 (m+2)}, \\quad l_m= 1-\\frac{3m-2}{m(m-1)(m+2)} +\\epsilon\n\\end{equation}\nis an exponent pair. This enables us to prove the following result. \n\n\\begin{corollary}\\label{cor2}\nFor each $a \\in [1\/2,1)$, the conclusion of Corollary \\ref{cor1} holds for some $b<1-a - a(1-a)^3  - 4.32 \\, a(1-a)^5$.\n\\end{corollary}\n\nThe value of $a$, for which $b(a,k_m,l_m)=b(a,k_{m+1},l_{m+1})$,  is given by\n$$\na_m := 1 -\\frac{1}{m} + \\frac{2-m^{-1}}{m^3+m^2+2m-1}\n$$\nIf $a>0.796...$ and $a\\in [a_{m-1},a_m]$, then $b$ is minimized by $b(a,k_m,l_m)$. \nThis yields slightly smaller values of $b$ than Corollary \\ref{cor2}.\n\n\\smallskip\n\\begin{table}[h]\n     \\begin{tabular}{ | c |  c |  c |  c | c | }\n    \\hline\n     $b$   & Interval for $a$ &  Exponent Pair \\\\ \\hline   \n     $517\/1648 +\\epsilon$ & $[0.500..., 0.579...]$ &   \\\\ \\hline\n    $ (110-55a)\/249+\\epsilon$ & $[0.579..., 0.590...]$ &  $BA(\\kappa,\\lambda)$ \\\\ \\hline\n     $ (55-42a)\/97+\\epsilon$ & $[0.590..., 0.701...]$ &  $(\\kappa,\\lambda)$ \\\\ \\hline\n        $ (152-139a)\/207+\\epsilon$ & $[0.701..., 0.766...]$ &  $A(\\kappa,\\lambda)$ \\\\ \\hline\n          $ (359-346a)\/427+\\epsilon$ & $[0.766..., 0.796...]$ &  $AA(\\kappa,\\lambda)$ \\\\ \\hline\n          $  b(a,k_m,l_m)$ & $  [a_{m-1},a_m], \\ m\\ge 5$ &  $(k_m,l_m)$ \\\\ \\hline\n     \\end{tabular}        \n   \\medskip\n     \\caption{Admissible values of $b$, depending on $a$.}\\label{table1}\n\\end{table}\n\nThe values $a=1-1\/m$, where $m\\ge 2$  is an integer, may be of particular interest.\nHere we have $a_{m-1}< a=1-1\/m < a_m$ and\n$$ b = b(1-1\/m,k_m,l_m) = \\frac{(m-1) \\left(m^3+m^2-3 m+2\\right)}{m^2 \\left(m^3-3 m+4\\right)} +\\epsilon.\n$$ \n\nThe exponent-pairs conjecture states that $(k,l)=(\\epsilon, 1\/2+\\epsilon)$ is an exponent pair for every $\\epsilon>0$. \n\\begin{corollary}\\label{cor3}\n If $(\\epsilon, 1\/2+\\epsilon)$ is an exponent pair, then the conclusion of Corollary \\ref{cor1} holds with $b=(1-a)\/2+\\epsilon$\n for each $a \\in [1\/2,1]$.\n\\end{corollary}\n\n\n \\begin{figure}[h]\\label{fig1}\n\\begin{center}\n\\includegraphics[height=70mm,width=112mm]{sep432.pdf}\n\\caption{Admissible values of $b$ based on Table \\ref{table1} (solid);\n$b=1-a-a(1-a)^3  - 4.32 \\, a(1-a)^5$ (dotted) from Cor.\\ \\ref{cor2};  \n$b=1-a$ (dashed) from the exponent pair $(k,l)=(0,1)$; \n $b=\\frac{1}{2}(1-a)$ (dot-dashed) from the exponent-pairs conjecture.}\n\\label{figure2}\n\\end{center}\n\\end{figure}\n\nIf one is only concerned with the existence of a single $y$-smooth number \nin short intervals, then a construction due to Friedlander and Lagarias \\cite{FL} \n(consider integers of the form $m^2-h^2=(m-h)(m+h)$, where $m=\\lceil \\sqrt{x}\\rceil$ \nand $h=0,1,2,\\ldots$) and an easy exercise (aided by a computer to deal with small values of $x$) lead to\nthe explicit estimate\n$$\n\\Psi\\bigl(x,\\sqrt{2x}\\bigr)  - \\Psi\\bigl(x-3 x^{1\/4},\\sqrt{2x}\\bigr)  \\ge 1 \\qquad (x\\ge 1).\n$$\nFrom Table \\ref{table1}, we find that our intervals are wider than $3x^{1\/4}$ \nwhen $a< 401\/556=0.721...$, but are shorter when $a>401\/556$. \n\n\\smallskip\n\n\\section{Proofs}\n\nLet $\\tau(n)$ be the number of positive divisors of $n$.\nThe following estimate is a special case of Theorem 2 of Shiu \\cite{Shiu}.\n\\begin{lemma}\\label{ShiuLem}\nLet $\\epsilon>0$ and $u \\in \\mathbb{R}$ be fixed. For $x \\ge 2$ and $x^\\epsilon \\le z \\le x$, we have\n$$\n\\sum_{x-z\\le n \\le x} (\\tau (n))^u \\ll z (\\log x)^{2^u -1}.\n$$\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm1}]\nLet $P(n)$ denote the largest prime factor of $n$.\nNote that the result holds if $z>x\/2$, so we may assume $z\\le x\/2$. \nDefine\n\\begin{equation}\\label{Sdef}\nS := \\sum_{x\/y\\le d \\le 2x\/y} \\sum_{x-z<n\\le x \\atop n \\equiv 0 \\bmod d} 1.\n\\end{equation}\nWe have\n\\begin{equation*}\n\\begin{split}\nS&  =\\sum_{x\/y\\le d \\le 2 x\/y} \\left( \\lfloor x\/d \\rfloor -  \\lfloor (x-z)\/d \\rfloor \\right) \\\\\n& =\\sum_{x\/y\\le d \\le 2 x\/y} z\/d - \\sum_{x\/y\\le d \\le 2x\/y}  \\psi(x\/d) + \\sum_{x\/y\\le d \\le 2x\/y}  \\psi((x-z)\/d) \\\\\n&\\ge z\/3+O\\bigl(\\min\\bigl(x^\\theta,x^{l\/(k+1)}y^{(k-l)\/(k+1)}\\bigr)\\bigr)  \\\\\n& \\ge z\/4,\n\\end{split}\n\\end{equation*}\nby \\eqref{UB} and the assumptions of Theorem \\ref{thm1}.\n \nNote that $y\\ge \\sqrt{2x}$ implies $2x\/y \\le y$. \nEvery $n$ counted in the inner sum of \\eqref{Sdef} has a divisor $d \\in [x\/y,2x\/y]\\subseteq [x\/y,y]$. \nSince $d\\le y$ and $n\/d\\le x\/(x\/y)=y$, we have $P(n)\\le y$, i.e. $n$ is $y$-smooth.  \nMoreover, each $n$ is counted at most $\\tau(n)$ times, once for each divisor $d$ of $n$ with $d\\in [x\/y,2x\/y]$. \nThus,\n$$\nS \\le \\sum_{x-z<n\\le x \\atop P(n)\\le y} \\tau(n).\n$$\nFor real numbers $t , u>1$ with $1\/t +1\/u=1$, H\\\"older's inequality yields\n\\begin{equation*}\n\\begin{split}\nS & \\le \\Biggl(\\sum_{x-z<n\\le x \\atop P(n)\\le y} 1\\Biggr)^{1\/t}  \\Biggl(\\sum_{x-z<n\\le x }\\tau(n)^u \\Biggr)^{1\/u} \\\\\n& \\ll \\left(\\Psi(x,y)-\\Psi(x-z,y)\\right)^{1\/t}  z^{1\/u} (\\log x)^{(2^u-1)\/u},\n\\end{split}\n\\end{equation*}\nby Lemma \\ref{ShiuLem}.\nSince $S\\ge z\/4$, we get\n$$\n\\Psi(x,y)-\\Psi(x-z,y) \\gg \\frac{z}{(\\log x)^{(2^u-1)\/(u-1)}}.\n$$\nThe last exponent has a minimum value of $\\nu=2.9882...$ at $u=2.1080...$\n\\end{proof}\n\n\\smallskip\n\n\\begin{proof}[Proof of Corollary \\ref{cor2}]\nFor $m\\ge 3$ and $a\\in [a_{m-1},a_m]$, we want to show that $b(a,k_m,l_m) < f(a)$, where $f(a)=1-a-a(1-a)^3-4.32a(1-a)^5$. \nSince $f''(a)<0$ for $1\/2<a<1$ and $b(a,k_m,l_m)$ is a linear function in $a$ for each $m$,\nit suffices to verify the inequality at the endpoints $a=a_m$. That is, we need to show that $b(a_m,k_m,l_m) < f(a_m)$\nfor $m\\ge 2$. We find that $f(a_m)-b(a_m,k_m,l_m) $ is a rational function in $m$ that is positive for every $m\\ge 1$. \nThis proves the claim for $a\\ge a_2 = 3\/5$. \nIf $1\/2 \\le a < 3\/5$, the result follows from Table \\ref{table1}. \n\\end{proof}\n\n\\section{Application to practical numbers}\n\nLet $\\mathcal{A}$ be the set of positive integers containing $n=1$ and all those $n \\ge 2$ with prime factorization $n=p_1^{\\alpha_1} \\cdots p_k^{\\alpha_k}$, $p_1<p_2<\\ldots < p_k$, which satisfy $p_1=2$ and \n\\begin{equation*}\np_{i} \\le  p_1^{\\alpha_1} \\cdots p_{i-1}^{\\alpha_{i-1}} \\qquad (2\\le i \\le k).\n\\end{equation*}\n\nThe significance of the set $\\mathcal{A}$ is that it is a subset of several notable integer sequences,\nincluding the practical numbers (i.e. integers $n$ such that every natural number $m\\le n$ can be expressed as a sum of distinct positive divisors of $n$ \\cite{PW, Saias,Ten86,PDD}), the $t$-dense numbers, for every $t\\ge 2$, (i.e. the ratios \nof consecutive divisors of $n$ are at most $t$, \\cite{PW, Saias,Ten86, PDD}), and the $\\varphi$-practical numbers (i.e. $x^n-1$ has a divisor in $\\mathbb{Z}[x]$ of every degree up to $n$, \\cite{PTW}).\n\nLet $\\nu=2.9882...$ be as in Theorem \\ref{thm1}, $C=(1-e^{-\\gamma})^{-1}=2.280...$, where $\\gamma=0.5772...$ is\nEuler's constant, and \n$$\\mu_0 := 2\\nu+2+C\\log 2 = 9.5569...$$ \n\\begin{theorem}\\label{thm2}\n Let $(k,l)$ be an exponent pair, $\\displaystyle \\beta =\\frac{5k+l+2}{6(k+1)}$ and $\\mu >\\mu_0$.\nThere exists a constant $K$ such that for $x\\ge z \\ge K x^\\beta$, the interval $[x-z,x]$ contains \n$\\gg z (\\log x)^{-\\mu}$  members of $\\mathcal{A}$. \n\\end{theorem}\n\nThe exponent pair $(k,l)=(13\/194+\\epsilon,76\/97+\\epsilon)=A(\\kappa,\\lambda)$ yields:\n\\begin{corollary}\\label{cor4}\nFor every $\\beta >605\/1242 = 0.4871...$  and $\\mu >\\mu_0$, the conclusion of Theorem \\ref{thm2} holds. \nAssuming the exponent-pairs conjecture,  it holds for every $\\beta > 5\/12=0.4166...$. \n\\end{corollary}\n\n\\begin{corollary}\\label{cor5}\nThe interval $[x-x^{0.4872},x]$ contains at least $x^{0.4872}(\\log x)^{-9.557}$ members of \n$\\mathcal{A}$, for all sufficiently large $x$.\n\\end{corollary}\n\nA quick search on a computer suggests that Corollary \\ref{cor5} probably holds for all $x\\ge 504$. \n\nIt is clear that Theorem \\ref{thm2} and its corollaries  remain valid if $\\mathcal{A}$ is replaced by any superset of $\\mathcal{A}$.\nIn the case of practical numbers, Corollary \\ref{cor5} improves on two earlier results: \nHausman and Shapiro \\cite{HS} found that the interval $[x^2, (x+1)^2]$ contains a practical  number for every $x\\ge 1$,\nin analogy with Legendre's conjecture for primes. \n Melfi \\cite[Thm. 9]{Mel95} sharpened this by showing that the \n interval $[x,x+K\\sqrt{x\/\\log\\log x}]$ contains a practical number for all large $x$ and some constant $K$.\n  \n Granville \\cite[Conj. 4.4.2]{Gran} states the conjecture that for every fixed $\\epsilon >0$,\nthe interval $[x-x^\\epsilon,x]$ contains a $x^\\epsilon$-smooth number for all $x\\ge x_0(\\epsilon)$. \nPomerance \\cite{PTalk} points out that this would imply the existence of a practical number \n(or member of $\\mathcal{A}$) in every interval $[x-x^\\epsilon,x]$ for large $x$. \n\nThe following observation follows at once from the definition of the set $\\mathcal{A}$.\n\n\\begin{lemma}\\label{ML}\nIf $n \\in \\mathcal{A}$ and $P(m)\\le n$, then $mn \\in \\mathcal{A}$. \n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm2}]\nIf $z> x\/2$, the result follows from Theorem 1.2 of \\cite{PDD}, so we may assume $z\\le x\/2$. \nLet $a=3\/4$. We have $\\displaystyle b=\\frac{3k+l}{4(k+1)}>0$, according to \\eqref{bdef}, and $\\beta = 1\/3+(2\/3)b>1\/3$. \n\nTheorem 1.2 of \\cite{PDD} shows that the number of $n \\in  \\mathcal{A} \\cap  (2x^{1\/3}, 3x^{1\/3}]$ is $ \\sim c x^{1\/3}\/\\log x$\nfor some positive constant $c$. \nLet $\\epsilon>0$ and $C=(1-e^{-\\gamma})^{-1}=2.280...$. \nBy Corollary 1 of \\cite{OMDD}, the number of these $n$ with $\\Omega(n)>(C+\\epsilon)\\log\\log n$ is $o(x^{1\/3}\/\\log x)$, so we may exclude such $n$ and assume\n$\\Omega(n)\\le (C+\\epsilon)\\log\\log n$. \n\nSince $n \\in (2x^{1\/3}, 3x^{1\/3}]$, the condition $z\\ge 3K x^\\beta$ implies $z\/n \\ge K (x\/n)^b$. \nBy Corollary \\ref{cor1}, for each of these $n$, \nthe interval $I_n:=[x\/n - z\/n, x\/n]$\ncontains $\\gg (z\/n)(\\log x\/n)^{-\\nu} \\gg z x^{-1\/3}(\\log x)^{-\\nu}$ integers $m$ that are $(x\/n)^{3\/4}$-smooth. \nNote that $mn \\in [x-z,x]$ for $m\\in I_n$. \n\n\nWe will show that for each of these pairs $(n,m)$ as described above, we have $mn \\in \\mathcal{A}$. \nLet $p = P(m)$. Since $n\\ge 2x^{1\/3}$, $p \\le (x\/n)^{3\/4} \\le x^{1\/2}2^{-3\/4}$. \nIf $p\\le x^{1\/3}$, then $mn \\in \\mathcal{A}$, by Lemma \\ref{ML}.\nIf $p> x^{1\/3}$, write $m=pr$ and note that $r=m\/p \\le x\/(np) < x^{1\/3}$. \nThus, $rn \\in \\mathcal{A}$ by Lemma \\ref{ML}. \nSince $p\\le x^{1\/2}2^{-3\/4}$, we have \n$ p^2 \\le x 2^{-3\/2}  < mn = pr n$ and hence $p < rn$.\nThus, $mn =prn \\in \\mathcal{A}$ also holds in this case, by Lemma \\ref{ML}. \n\nThe number of pairs $(m,n)$ is $\\gg z(\\log x)^{-1-\\nu}$,\nbut several pairs may lead to the same product $mn$. \nWe have $\\tau(n)\\le 2^{\\Omega(n)} \\le (\\log x)^{C \\log 2 + \\epsilon}$.\nBy Lemma \\ref{ShiuLem}, we have $\\sum_{m\\in I_n} \\tau(m) \\ll (z\/n)\\log x$.\nSince the number of $m\\in I_n$ that are $(x\/n)^{3\/4}$-smooth is $ \\gg (z\/n) (\\log x)^{-\\nu}$,\nwe have $\\tau(m) \\ll (\\log x)^{\\nu+1}$ for a positive proportion of them.\nThus, we may assume $\\tau(m) \\ll (\\log x)^{\\nu+1}$, and therefore $\\tau(mn)\\ll (\\log x)^{\\nu+1+C\\log 2 +\\epsilon}$.\nIt follows that the number of distinct products $mn$ is \n$$\n\\gg  \\frac{z (\\log x)^{-1-\\nu}}{ (\\log x)^{\\nu+1+C\\log 2 +\\epsilon}}\n=\\frac{z}{(\\log x)^{\\mu_0 + \\epsilon}}.\n$$\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}