data_all_eng_slimpj/shuffled/split2/finalzzaofd

{"text":"\\section{Introduction}\n\n\n\nAt high velocities space and time are merged together into Minkowski\nspacetime, and even though both distances and durations depend on the\nobservers frame, then the 4-vector $(ct, \\vec x)$ has invariant length\nunder Lorentz transformations.  Similarly the energy-momentum\n4-vector, $(\\varepsilon\/c , \\vec p)$, is a proper 4-vector in\nMinkowski space. The classical conservation laws, like energy and\nmomentum conservation arising from symmetries in time and space, thus\nhave related conservation laws in relativistic physics.\n\nHowever, for other objects, such as the (polar vector) dynamic mass\nmoment, $\\vec N =  ct \\vec p -\\varepsilon \\vec x\/c $, or the (axial\nvector) angular momentum, $\\vec L = \\vec x \\times \\vec p$, the\ncorresponding relativistic conservation laws are often not discussed\nin detail. One reason for this is that there is no way of combining\n$\\vec N$ and $\\vec L$ into a proper 4-vector.  Instead, one must\ncombine $\\vec N$ and $\\vec L$ into an anti-symmetri rank-2 tensor,\n$M^{\\mu \\nu}$ \\citep{landau}.\n\nA very similar issue is well-known from electromagnetisme, where the\n(polar vector) electric field, $\\vec E$, and the (axial vector)\nmagnetic field, $\\vec B$, also combine into an antisymmetri rank-2\ntensor, $F^{\\mu \\nu}$.  When learning about electromagnetism this is\nfrustrating to some, since most of our intuition is based on the\nfields $(\\vec E, \\vec B)$, however, when performing a Lorentz\ntransformation, one most often finds oneself performing the\ncalculations with the physically somewhat less transparant tensor\n$F^{\\mu \\nu}$.\n\nWhile contemplating the physical meaning of the mathematical space\nwhere $F^{\\mu \\nu}$ and $M^{\\mu \\nu}$ live, one is naturally drawn to\nspacetime algebra (STA) \\citep{hestenes1966, 2003AmJPh..71..691H},\nwhich provides a geometric explanation for the connection between\ntensors like $F^{\\mu \\nu}$ and $M^{\\mu \\nu}$ and Minkowski space.\n\n\n\nIn the sufficiently mature scientific field of STA, it is well-known\nhow the electromagnetic bi-vector field, ${\\bf F}$, naturally is used\nto derive both Maxwell's equations and the Lorentz force. Given the\nstrong mathematical similarities between the ``electromagnetic''\nbi-vector ${\\bf F}$ and the ``mechanics'' bi-vector\n${\\bf M}$, it appears natural to derive the equations and forces\nwhich are dictated by ${\\bf M}$, in particular since the\nstructure of STA uniquely defines these equations and\nforces.\n\n\nBelow the same methods are applied to the bi-vector field, ${\\bf M}$,\nas have previously been applied to ${\\bf F}$ in electromagnetism, and\nit is shown how new forces appear naturally from the bi-vector field\n${\\bf M}$.  One of these inverse distance-squared force-terms depends on the\ninternal velocity dispersion of an object (which for instance could be\na distant galaxy). It is speculated to which degree this new force possibly may\nbe related to the force which was recently suggested as an explanation\nfor the observed acceleration of the\nuniverse~\\citep{2021ApJ...910...98L}.\n\n\\section{Spacetime algebra}\n\nSpacetime algebra starts with Minkowski space, ${\\cal M}_{1,3}$, with\nthe metric signature $(+,-,-,-)$, and a chosen basis $\\{\\gamma_\\mu\n\\}_{\\mu=0}^3$ of ${\\cal M}_{1,3}$. These 4 orthonormal vectors are the\nbasis for 1-blades. The 2-blade elements are the 6 antisymmetric\nproducts $\\gamma _{\\mu \\nu} \\equiv \\gamma_{\\mu} \\gamma_{\\nu} $. The\nproduct is here given by the sum of the dot and wedge product: $a b =\na \\cdot b + a \\wedge b$ \\citep{hestenes2015,doranlasenby}.\nThe wedge operator, $\\wedge$, is the 4-dimensional generalization\nof the 3-dimensional cross-product.\nContinuing over 3-blades, $\\gamma _{\\mu\n  \\nu \\delta}$, one finally reaches the highest grade, the\npseudoscalar $I \\equiv \\gamma _0 \\gamma _1 \\gamma _2 \\gamma _3$, which\nrepresents the unit 4-volume in any basis. Interestingly one has $I^2\n= -1$.\n\nFor the discussion below, the bi-vectors are  important:\nthese are oriented plane segments, and examples include the\nelectromagnetic field $ {\\bf F} = \\vec E + \\vec B \\, I$, and the\nangular momentum ${\\bf M} = x \\wedge p$, where $x$ and $p$ are proper\n4-vectors~\\citep{hestenes2015,doranlasenby}.\nFrom a notational point of view vector-arrows are used above spatial 3-vectors\nlike $\\vec E$ or $\\vec p$, no-vector-arrows are used for proper 4-vectors like\n$x$ and $w$, and boldface is used for bi-vectors like ${\\bf F}$ and\n${\\bf M}$.\n\n\n\n\n\\section{Electromagnetism}\n\nThe case of electromagnetism in STA is  well described in the\nliterature \\citep{hestenes2015, dressel2015}, and serves as a starting\npoint here.  The 4 Maxwells equations can be written\n\\begin{equation}\n  \\nabla {\\bf F} = j \\,\n  \\label{eq:maxwell}\n\\end{equation}\nwhere the complex current may contain both eletric (vector) and\nmagnetic (trivector) parts, $j_e + j_m I$. The bi-vector is given by\n${\\bf F} = \\vec E + \\vec B I$, and the derivative $\\nabla {\\bf F} = \\nabla\n\\cdot {\\bf F} + \\nabla \\wedge {\\bf F}$ produces both a vector and a\ntrivector field.\n\nUsing the time-direction, $\\gamma_0$, one can decompose the derivative along\na direction parallel to and perpendicular to $\\gamma_0$,\n$\\nabla = \\left( \\partial _0 - \\vec \\nabla \\right) \\gamma_0 $,\nwhere $ \\vec \\nabla $ is the frame-dependent relative 3-vector derivative.\nIt is now straight forward to expand eq.~(\\ref{eq:maxwell}) to the\n4 Maxwells equation~\\citep{hestenes2015, dressel2015}.\n\nIt is important to stress, that eq.~(\\ref{eq:maxwell}) is not only a\nmatter of compact notation, it is indeed the only logical extension\nbeyond the most trivial equation in STA, $\\nabla {\\bf F} = 0$. The\nonly thing missing is to connect the bi-vector field to observables:\nthis is done through observations, which also establish the units of\n$\\vec E$ and $\\vec B$.\n\n\n\\subsection{The Lorentz force}\nThe classical Lorentz force is given by\n\\begin{equation}\n  \\frac{d\\vec p}{d t} = q \\left(  c \\vec E + \\vec v \\times \\vec B \\right) \\, ,\n  \\label{eq:lorentz}\n\\end{equation}\nwhich effectively arose as a clever guess to explain observations.\nThe cross, $\\times$, refers to the normal 3-dimensional cross-product.\nIn\nthe standard Euler-Lagrange formalism the Lorentz force appears when\nadding a term to the Lagrangian, $q w_\\mu A^\\mu$, where $A$ is the\n4-vector potential.\n\nIn STA the Lorentz force appears when one contract the bi-vector field\n${\\bf F}$ with a proper 4-vector velocity $w$. Using that the 4-vector\n$w$ is connected with the para-vector, $w_0+\\vec w$, via\n$\\gamma_0$~\\citep{doranlasenby}, namely~\\footnote{The\nright-multipliation by the timelike vector $\\gamma_0$ isolates the\nrelative quantities of that frame~\\citep{dressel2015}, e.g. $x\n\\gamma_0 = \\left( ct + \\vec x \\right)$.}  $w = \\left( w_0 + \\vec w\n\\right) \\gamma_0$, one gets\n\\begin{equation}\n  \\left(  {\\bf F} \\cdot w \\right) q \\frac{d\\tau}{dt} \\gamma_ 0=\n  q  \\vec E \\cdot \\vec v +\n  q \\left( c \\vec E + \\vec v \\times \\vec B\\right) \\, ,\n\\label{eq:lorentzforce}\n\\end{equation}\nwhere the first term on the r.h.s. is the rate of work,\n$d\\varepsilon\/d(ct)$, we use $\\vec w = \\gamma \\vec v$,\nand the last parenthesis on the r.h.s is exactly\nthe Lorentz force in eq.~(\\ref{eq:lorentz}).\n\n\nIf the current was complex\nthere could be another force term allowed~\\citep{dressel2015}, namely\n\\begin{equation}\n  {\\bf F} \\cdot \\left( w I \\right) \\gamma_0 = \\left( \\left( {\\bf F} \\wedge w \\right) I \\right) \\gamma_0\n    = \\vec B \\cdot \\vec v + \\left( c \\vec B - \\vec v \\times \\vec E \\right) \\, .\n\\end{equation}\n\nTo summarize, the full 4 Maxwells equations appear naturally from the\ngeometric structure of STA, through the equation $\\nabla {\\bf F} =\nj$. The Lorentz force also appears naturally in STA when contracting\nthe bi-vector field, ${\\bf F}$ with the 4-vector current, $dp\/d\\tau =\n{\\bf F} \\cdot (qw)$, where $p$ is the proper energy-momentum 4-vector.\n\n\\section{Angular momentum}\nIn order to generalize the 3-dimensional angular momentum, $\\vec L =\n\\vec x \\times \\vec p$, one uses the proper 4-dimensional $x=(ct +\\vec\nx)\\gamma_0$ and $p= (\\varepsilon\/c + \\vec p) \\gamma_0$, to create the\nbi-vector ${\\bf M}$~\\citep{landau, dressel2015}\n\\begin{eqnarray}\n  {\\bf M} &=& x \\wedge p  \\nonumber \\\\\n  &=& \\frac{\\varepsilon \\vec x}{c} - ct \\vec p - \\vec x \\times \\vec p I  \\nonumber \\\\\n  &=& - \\vec N - \\vec L I  \\, .\n\\end{eqnarray}\nOnly ${\\bf M}$ is a proper geometric object, and the split into\ndynamic mass moment and angular momentum requires that one specifies\n$\\gamma_0$, in exactly the same way that ${\\bf F}$ is the proper\ngeometric object of electromagnetism, and the separation into $\\vec E$\nand $\\vec B$ fields requires specification of a frame by the choice of\n$\\gamma_0$.\n\n\nIt is now clear how everything can be repeated from the case of\nelectromagnetism: where one had a bi-vector ${\\bf F}$ and relative\n3-vectors $\\vec E$ (polar) and $\\vec B$ (axial), then one now has a\nbi-vector ${\\bf M}$ and relative 3-vectors $- \\vec N$ (polar) and\n$-\\vec L$ (axial). The signs could have been defined away, but are kept\nto agree with the standard notation in the literature~\\citep{landau}.  When\nderiving the Lorentz force for electromagnetism, by dotting the\nbi-vector field ${\\bf F}$ with a charge-current, $q w$, one needs\nexperimental data to get the units right ($\\epsilon_0$ and $\\mu_0$ for\nthe E- and B-fields, respectively)~\\citep{hestenes2015}. In a similar\nfashion experimental data is needed to get the units for a force\ndefined by dotting the field ${\\bf M}$ with a ``mass-current'', $m_t w$.\n\n\nThe simplest possible equation describing the evolution of the\nbi-vector field is specified by the structure of STA, namely\n\\begin{equation}\n\\nabla {\\bf M} = j_{\\bf M} \\, .\n\\end{equation}\nThis Letter is not focusing on the details of the source on the r.h.s. (which could be zero),\nhowever, for the sake of generality it is allowed to contain both a\nvector and a trivector term $ j_{\\bf M} = j_{\\bf 1} + j_{\\bf 3} I$.\nThe resulting equations split into two equations for the relative\nscalars\n\\begin{eqnarray}\n-\\vec \\nabla \\cdot \\vec N &=& \\rho_1  \\, , \\\\\n\\label{eq:relativescalars} \n- \\vec \\nabla \\cdot \\vec L &=& \\rho_3 \\, ,\n\\label{eq:relativescalars2}\n\\end{eqnarray}\n(where $\\rho_i$ refer to the 0-component of the sources) and two\nequations for the relative 3-vectors, just like Maxwell's equations\ndid.\n\\begin{eqnarray}\n  -\\partial_0 \\vec N + \\vec \\nabla \\times \\vec L &=& - \\vec J_1 \\, ,\\\\\n  \\partial_0 \\vec L + \\vec \\nabla \\times \\vec N &=&  \\vec J_3  \\, ,\n  \\label{eq:relvec2}\n\\end{eqnarray}\nwhere $\\vec J_i$ refer to the 3 spatial components of the sources.\nThe details of these 4 equations\nwill be discussed elsewhere~\\citep{students}.\n\n\nInstead, the force which appears from the\ncontraction with a mass-current, $m_t \\omega$, where $w$ again is a\nproper 4-velocity, and $m_t$ is the inertia of the test particle,\nwill now be calculated.  From the term ${\\bf M} \\cdot w$ one gets\n\\begin{equation}\n  \\left( {\\bf M} \\cdot w  \\right) \\frac{d\\tau}{dt} \\gamma_0 = -\\vec N \\cdot \\frac{\\vec v}{c} +\n  \\left( -  \\vec N - \\frac{\\vec v}{c} \\times \\vec L \\right) \\,.\n  \\label{eq:newforce}\n\\end{equation}\nThe first term on the r.h.s. is similar to a rate of work. However,\nthe last parenthesis of eq.~(\\ref{eq:newforce})\ncontains the new forces of interest here,\nand\nwill be discussed in section \\ref{sec:newforce}\nbelow.  One could also have considered a force arising from $\\left(\n{\\bf M} \\wedge w \\right) I$, which looks like\n\\begin{equation}\n\\left( {\\bf M} \\wedge w \\right) I \\frac{d\\tau}{dt} \\gamma_0 =\n   - \\vec L \\cdot \\frac{\\vec v}{c}\n  + \\left( -\\vec L + \\frac{\\vec v}{c} \\times \\vec N \\right) \\, ,\n  \\label{eq:newforce2}\n\\end{equation}\nhowever, it is left for a future analysis to study this.\n\n\n\\section{The new force terms}\n\\label{sec:newforce}\n\nLet us consider a collection of particles at a large distance, $\\vec\nr_0$.  The particles may have different inertia, $m_i$, but move\ncollectively with an average velocity, $\\vec V$. If one considers\nparticles in a cosmological setting, then the velocity is a\ncombination of the Hubble expansion and peculiar velocity, $\\vec V = H\n\\vec r_0 + \\vec v_p$, and at large distances the peculiar velocity is\nsubdominant. The collection of particles may have internal motion,\nwhich is simplified with an internal velocity dispersion,\n$\\sigma^2$. Practically when calculating the velocity\ndispersion there will be terms including both the Hubble expansion,\n$v_H = H r$, and also the background density of both matter and the\ncosmological constant, however, these terms happen to exactly cancel\neach other~\\citep{2013MNRAS.431L...6F}, and one can therefore\ncalculate $\\sigma^2$ as if the structure is alone in a non-expanding\nuniverse.\n\nThe dynamic mass moment is given by\n\\begin{equation}\n\\vec N = \\sum \\left( ct \\vec p_i - \\frac{\\varepsilon_i \\vec r_i}{c} \\right) \\, ,\n\\end{equation}\nwhere the sum is over all particles involved~\\citep{landau}. If one\ndivides both terms by the total energy, $\\varepsilon_{tot} = \\sum\n\\varepsilon_i$, then one gets\n\\begin{equation}\n  \\frac{\\vec N }{\\varepsilon_{tot}} = \\left( \\frac{ct \\sum \\vec\n    p_i}{\\sum \\varepsilon_i} - \\frac{\\sum \\varepsilon_i \\vec r_i}{c \\sum\n    \\varepsilon_i} \\right) \\, ,\n\\end{equation}\nThe first term is just $ct$ times the average velocity. At small velocities\none has $\\varepsilon \\approx m_ic^2$ and hence the last term describes\nthe relativistic center of inertia, $\\vec R_{cm} = \\sum (m_i \\vec\nr_i)\/ \\sum m_i$.\nIf the centre of inerti moves at constant velocity (now ignoring sums over\nparticles), then one has $\\vec r = \\vec r_0 + \\vec V t$, and hence\n\\begin{equation}\n  \\vec N = - m c \\vec r_0 \\,  .\n\\label{eq:Nr}\n\\end{equation}\n\n\nWhen considering the Lorentz force in eq.~(\\ref{eq:lorentzforce}) one needs\nto get the units right to get \n$\\vec E = q \\vec r \/(4 \\pi \\epsilon_0 |r|^3)$, which includes\nthe observable vacuum permittivity, $\\epsilon_0 $, and also Coulombs\ninverse distance-square law (resulting from Gauss' and Faradays' laws combined).\nEffectively this means dividing by $\\epsilon_0 |r|^2$.\n\n\nThe new force terms in the parenthesis of eq.~(\\ref{eq:newforce}) will\nnow be considered.  Since the mass-current is related to gravity, one\nshould multiply by Newtons gravitational constant, $G = 6.67 \\times\n10^{-11} {\\rm m}^3\/({\\rm s}^2 \\, {\\rm kg})$.  To get a well-behaved\nfield one divides by distance to power 3, which will lead to an inverse\ndistance-square force: this comes from the integral over\neq.~(\\ref{eq:relativescalars}), using the sphericity from\neq.~(\\ref{eq:relvec2}) with $\\vec J_3=0$ and the expression in\neq.~(\\ref{eq:Nr}). This last point is easily recognized by considering\nthe change of notation, $\\vec N \\rightarrow \\vec g \/(4 \\pi G)$, and\n$\\rho_1 \\rightarrow \\rho_m$ which is the mass density, which means\nthat eq.~(\\ref{eq:relativescalars}) is written as $\\vec \\nabla \\cdot\n\\vec g = - 4 \\pi G \\rho_m$. This equation is clearly regognized as\nleading to Newtons gravitational law.  Finally to get the units right,\nit is divided by $c$.\n\n\n\nThis implies that one has a force-term that looks like\n\\begin{equation}\n  - \\kappa \\, \\frac{G m_t m \\vec r_0} {| \\vec r_0 |^3} \\, .\n\\label{eq:newton}\n\\end{equation}\nwhere $\\kappa$ is\nan unknown, dimensionless number, which must be\ndetermined from observations,\nand $m_t$ is from the\nmass-current, $m_t w$.\nIn the case of $\\kappa = 1$ this is\njust Newtons gravitational force.\nIn the above picture, it thus appears that the Newtonian gravity may\nbe interpreted as a gravitational analogue to the Coulomb force from\nelectromagnetism. Since the masses are always positive, the\ngravitational force is always attractive.\n\n\n\nWhen the structure under consideration contains a dynamical term\nproportional to the velocity dispersion, $\\sigma^2$, which for\ninstance can arise in a dwarf galaxy where the stars and dark matter\nparticles are orbiting in the local gravitational potential, then the\npotential will be minus 2 times the kinetic energy according to the\nvirial theorem~\\citep{bt2}, $2T+U=0$, and hence one writes the\nenergy as\n\\begin{equation}\n\\varepsilon_i = m_ic^2 - \\frac{1}{2} m_i \\sigma_i^2 \\, .\n\\end{equation}\nIn this case one ends up with a new force of the form \n\\begin{equation}\n   \\frac{\\tilde \\kappa}{2} \\, \\frac{\\sigma^2}{c^2} \\,\n    \\frac{m_tm_iG \\vec r_0}{\\| \\vec r_0 \\|^3} \\, ,\n\\label{eq:sigma2}\n\\end{equation}\nwhere the dispersion has been  normalized to $c$.  This force is always\nrepulsive. In the case of $\\tilde \\kappa =1$ this is just a minor\ncorrection to the normal gravitational attraction, e.g. for galaxy\nclusters with velocity dispersions of $1000$ km\/sec this is a\n$10^{-5}$ correction, and for dwarf galaxies much less. Possibly for\nmotion near very compact objects, this correction may eventually be\nobservable.\n\n\n\n\nVelocity-dependent forces are well-known, including the Coriolis-force\nand the Lorentz-force, however, this is, to our knowledge, the first\nderived long-distance force depending on velocity squared.  In an\natttempt to make kinetic energy depend on relative velocities (rather\nthan absolute velocities) similarly to how potential energy depends on\nrelative position, Schr\\\"odinger suggested a new gravitational force\nproportional to velocity squared \\citep{1925AnP...382..325S}. His\nforce has essentially the same form as the force derived above,\nhowever, its existence was postulated on rather philosofical grounds.\n\n\n\nA recent paper demonstrated that a universe which contains no dark\nenergy, but instead includes a new repulsive inverse  distance-square force\nproportional to internal velocity dispersion squared, just like\nequation~(\\ref{eq:sigma2}), could have an accelerated expansion which\nfairly closely mimics the accelerated expansion induced by the\ncosmological constant~\\citep{2021ApJ...910...98L}.  In that paper it\nwas implicitely suggested that such a force might conceivably exist amongst the\ndark matter particles. What has been shown in this Letter is, that\nsuch a force indeed may exist, and that it is not specifically related\nto the dark matter particle, but instead related to gravity in general.\nOne of the concerns with the suggestion discussed in\n\\cite{2021ApJ...910...98L} is the potential instability of\ncosmological structures, however, from the derivation above it is clear\nthat the new force derived here\ncomes from the internal dispersion (as opposed to relative velocities),\nand hence there is no instability concern.\n\n\nThe magnitude of the dimensionless $\\tilde \\kappa$ in\neq.~(\\ref{eq:sigma2}) is unknown in the present derivation, however,\nit should logically be unity.  The new force of the\npaper~\\citep{2021ApJ...910...98L} should have a numerical value of\n$\\tilde \\kappa \\sim 10^6-10^8$. From the derivation above there is no\nindication where such a large factor should come from.\n\n\n\n\n\n\n\n\\section{Conclusion}\nIt was recently suggested that if a force which depends on velocity\nsquared exists in Nature, then it may induce an effect on cosmological\nscales which mimics the accelerated expansion of the standard\ncosmological constant~\\citep{2021ApJ...910...98L}. The present Letter\ndemonstrates one such concrete possibility.  The derivation here is\nframed in the geo\\-metric structure of spacetime\nalgebra~\\citep{hestenes2015}, and takes as starting point the\nrelativistic generalization of angular momentum which includes the\ndynamic mass moment, $\\vec N = ct \\vec p - \\varepsilon \\vec\nr\/c$. Since the energy in this term, $\\varepsilon$, contains the\nvelocity dispersion of a distant cosmological object, then an inverse\ndistance-square force naturally appears, which is proportional to\n$\\sigma^2$.\nSuch a force \nmay lead to a slightly reduced\ngravitational force for extremely compact objects.\nThe magnitude of the force derived here is significantly\nsmaller than needed to explain the present day accelerating\nuniverse~\\citep{2021ApJ...910...98L}.\n\n\n\n\n\n\n\n\\section*{Acknowledgement}\nIt is a pleasure to thank Max Emil K.S. Sondergaard, Magnus B. Lyngby\nand Nicolai Asgreen for interesting discussions.  I thank Mario\nPasquato for bringing the 1925 Schr\\\"odinger paper to my attention.\n\n\\section{Data availability}\nNo new data were generated or analysed in support of this research.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Main Results}\n\\begin{thm}\nLet $\\varphi$ be an analytic self-map of the unit disk $\\mathbb{D}$ and $g\\in H(\\mathbb{D})$, $\\psi\\in\\mathcal{D}$. Then $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded if and only if $$\\sup_{z\\in\\mathbb{D}}e^{\\psi(\\varphi(z))-2\\psi(z)}\\sqrt{\\Delta\\psi(\\varphi(z))}|g'(z)|<\\infty.$$\n\\end{thm}\n\\begin{proof}\nBy Lemma 2.2, $$e^{\\psi(\\varphi(z))}\\sqrt{\\Delta\\psi(\\varphi(z))}\\sim \\sqrt{K(\\varphi(z),\\varphi(z))},$$\ntherefore, it suffices for us to prove that\n$V_{\\varphi}^{g}$ is bounded if and only if $$\\underset {z\\in\\mathbb{D}}{\\textup{sup}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|<\\infty.$$\n\nFirst, assume that $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded. For every $w\\in\\mathbb{D}$, let $f_{w}(z)=k_{w}(z)=\\frac{K(z,w)}{\\sqrt{K(w,w)}}.$ It is easy to check that $f_{w}\\in AL_{\\psi}^{2}(\\mathbb{D})$ and  $\\|f_{w}\\|_{L_{\\psi}^{2}}=1$ for any $w\\in\\mathbb{D}$.\n\nHence, for a fixed $w\\in \\mathbb{D}$,\n\\begin{eqnarray*}\n|f_w(\\varphi(z))||g'(z)|e^{-2\\psi(z)}\n&=&|(V_{\\varphi}^{g}f_{w})^{'}(z)|e^{-2\\psi(z)}\\\\\n&\\leq&\\|V_{\\varphi}^{g}f_{w}\\|_{B_{\\psi}}\\\\\n&\\leq&\\|V_{\\varphi}^{g}\\|\\|f_{w}\\|_{L_{\\psi}^{2}}\\\\\n&=&\\|V_{\\varphi}^{g}\\|< \\infty\n\\end{eqnarray*}\nfor any $z\\in \\mathbb{D}.$\nNoticing that\n$$f_w(\\varphi(z))=\\frac{K(\\varphi(z),w)}{\\sqrt{K(w,w)}},~~~~\\mbox{for all}~z\\in \\mathbb{D}, $$\nby setting $w=\\varphi(z)$, we have\n$$f_{\\varphi(z)}(\\varphi(z))=\\frac{K(\\varphi(z),\\varphi(z))}{\\sqrt{K(\\varphi(z),\\varphi(z))}}=\\sqrt{K(\\varphi(z),\\varphi(z))}.$$\nIt follows that\n $$\\underset {z\\in\\mathbb{D}}{\\textup{sup}}\\sqrt{K(\\varphi(z),\\varphi(z))}|g'(z)|e^{-2\\psi(z)}<\\infty.$$\n\nConversely, assume that $$\\underset {z\\in\\mathbb{D}}{\\textup{sup}}\\sqrt{K(\\varphi(z),\\varphi(z))}|g'(z)|e^{-2\\psi(z)}=M<\\infty.$$\nFor any $f\\in AL_{\\psi}^{2}(\\mathbb{D})$. By Lemma 2.3, we have\n\\begin{eqnarray*}\n\\|V_{\\varphi}^{g}f\\|_{B_{\\psi}}&=&|V_{\\varphi}^{g}f(0)|+\\underset {z\\in\\mathbb{D}}{\\textup{sup}}|(V_{\\varphi}^{g}f)'(z)|e^{-2\\psi(z)}\\\\\n&=&\\underset {z\\in\\mathbb{D}}{\\textup{sup}}|f(\\varphi(z))g'(z)|e^{-2\\psi(z)}\\\\\n&\\leq&\\underset {z\\in\\mathbb{D}}{\\textup{sup}}\\|f\\|_{L_{\\psi}^{2}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|\\\\\n&=&\\|f\\|_{L_{\\psi}^{2}}M.\n\\end{eqnarray*}\nTherefore, $V_{\\varphi}^{g}$ is bounded.\n\\end{proof}\n\n\n\n\n\n\\begin{thm}\nLet $\\varphi$ be an analytic self-map of the unit disk $\\mathbb{D}$ and $g\\in H(\\mathbb{D})$, $\\psi\\in\\mathcal{D}$. Then $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is bounded if and only if $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded and $$\\lim_{|z| \\rightarrow 1}e^{-2\\psi(z)}|g'(z)|=0.$$\n\\end{thm}\n\\begin{proof}\nAssume that $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is bounded. It is clear that\n$V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded. Taking $f(z)=1\\in AL_{\\psi}^{2}(\\mathbb{D})$ and $V_{\\varphi}^{g}f\\in B_{\\psi,0}(\\mathbb{D})$, then\n\\begin{eqnarray*}\n0=\\underset {|z|\\rightarrow1}{\\textup{lim}}|(V_{\\varphi}^{g}f)^{'}(z)|e^{-2\\psi(z)}&=&\\underset {|z|\\rightarrow1}{\\textup{lim}}|f(\\varphi(z))||g'(z)|e^{-2\\psi(z)}\\\\\n&=&\\underset {|z|\\rightarrow1}{\\textup{lim}}|g'(z)|e^{-2\\psi(z)}\n\\end{eqnarray*}\n\nConversely, suppose that $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded and $\\lim_{|z| \\rightarrow 1}e^{-2\\psi(z)}|g'(z)|=0$. For each polynomial $p(z)$, the following inequality holds\n\\begin{eqnarray*}\n|(V_{\\varphi}^{g}p)^{'}(z)|e^{-2\\psi(z)}&=&|p(\\varphi(z))||g'(z)|e^{-2\\psi(z)}\\\\\n&\\leq& M_{p}|g'(z)|e^{-2\\psi(z)}.\n\\end{eqnarray*}\nwhere $M_{p}=\\underset {z\\in\\mathbb{D}}{\\textup{sup}}|p(z)|$. Since $M_{p}<\\infty$ and $\\underset {|z|\\rightarrow1}{\\textup{lim}}|g'(z)|e^{-2\\psi(z)}=0$, then\n$$\\underset {|z|\\rightarrow1}{\\textup{lim}}|(V_{\\varphi}^{g}p)^{'}(z)|e^{-2\\psi(z)}=0.$$\nThat means for each polynomial $p$, $V_{\\varphi}^{g}p(z)\\in B_{\\psi,0}(\\mathbb{D})$. Since the set consisting of  polynomials is dense in $AL_{\\psi}^{2}(\\mathbb{D})$, for every $f\\in AL_{\\psi}^{2}(\\mathbb{D})$, there is a sequence of polynomials $\\{p_{k}\\}_{k\\in\\mathbb{N}}$ such that\n$$\\|f-p_{k}\\|_{L_{\\psi}^{2}}\\rightarrow 0~~~~(k\\rightarrow\\infty).$$\nHence,$$\\|V_{\\varphi}^{g}f-V_{\\varphi}^{g}p_{k}\\|_{B_{\\psi}}\\leq\\|V_{\\varphi}^{g}\\|\\|f-p_{k}\\|_{L_{\\psi}^{2}}\n\\rightarrow0(k\\rightarrow\\infty).$$\nSince $V_{\\varphi}^{g}p_{k}\\in B_{\\psi,0}(\\mathbb{D})$ and $B_{\\psi,0}(\\mathbb{D})$ is the the closed subset of $B_{\\psi}(\\mathbb{D})$,\nwe have $V_{\\varphi}^{g}(AL_{\\psi}^{2}(\\mathbb{D}))\\subset B_{\\psi,0}(\\mathbb{D})$.\\\\\nSince $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded , we see that $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is bounded.\n\\end{proof}\n\n\n\n\\begin{thm}\nLet $\\varphi$ be an analytic self-map of the unit disk $\\mathbb{D}$ and $g\\in H(\\mathbb{D})$, $\\psi\\in\\mathcal{D}$. Then $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is compact if and only if $$\\lim_{|\\varphi(z)|\\rightarrow 1}e^{\\psi(\\varphi(z))-2\\psi(z)}\\sqrt{\\Delta\\psi(\\varphi(z))}|g'(z)|=0.$$\n\\end{thm}\n\\begin{proof}\nBy Lemma 2.2, $$e^{\\psi(\\varphi(z))}\\sqrt{\\Delta\\psi(\\varphi(z))}\\sim \\sqrt{K(\\varphi(z),\\varphi(z))},$$\ntherefore, we should only show that $V_{\\varphi}^{g}$ is compact if and only if\n$$\\lim_{|\\varphi(z)|\\rightarrow 1}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|=0.$$\n\nFirst, assume that $\\lim_{|\\varphi(z)|\\rightarrow 1}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|=0$, then for any $\\varepsilon>0$, there is a positive real number $r_{0}\\in(0,1)$ such that $$\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|<\\varepsilon$$ when $r_{0}<|\\varphi(z)|<1$.\nBesides, $\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|$ is bounded when $|\\varphi(z)|\\leq r_{0}$, it is easy to see that\n$$\\underset {z\\in\\mathbb{D}}{\\textup{sup}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|<\\infty.$$\nBy Theorem 3.1,  $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded.\n\nLet $\\{f_{k}\\}_{k\\in\\mathbb{N}}$ be a bounded sequence in $AL_{\\psi}^{2}(\\mathbb{D})$ which uniformly converges to zero on any compact subset of $\\mathbb{D}$ as $k\\rightarrow\\infty$.\nAssume that for any $k\\in\\mathbb{N}$,$\\|f_{k}\\|\\leq C,$ for some positive constant $C$.\nNote that $$\\underset {|\\varphi(z)|\\leq r_{0}}{\\textup{sup}}|g'(z)|e^{-2\\psi(z)}\\leq M$$ for some constant $M$.\nIt follows that\n\\begin{eqnarray*}\n|(V_{\\varphi}^{g}f_{k})^{'}(z)|e^{-2\\psi(z)}&=&|f_{k}(\\varphi(z))||g'(z)|e^{-2\\psi(z)}\\\\\n&\\leq&\\underset {|\\varphi(z)|\\leq r_{0}}{\\textup{sup}}|f_{k}(\\varphi(z))||g'(z)|e^{-2\\psi(z)}\\\\\n&&+\n\\underset {|\\varphi(z)|> r_{0}}{\\textup{sup}}|f_{k}(\\varphi(z))||g'(z)|e^{-2\\psi(z)}\\\\\n&\\leq& M\\underset {|\\varphi(z)|\\leq r_{0}}{\\textup{sup}}|f_{k}(\\varphi(z))|\\\\\n&&+\n\\|f_{k}\\|_{L_{\\psi}^{2}}\\underset {|\\varphi(z)|> r_{0}}{\\textup{sup}}\\sqrt{K(\\varphi(z),\\varphi(z))}|g'(z)|e^{-2\\psi(z)}\\\\\n&\\leq& M\\underset {|\\varphi(z)|\\leq r_{0}}{\\textup{sup}}|f_{k}(\\varphi(z))|+\\varepsilon\\|f_{k}\\|_{L_{\\psi}^{2}}\\\\\n&\\leq& M\\underset {|\\varphi(z)|\\leq r_{0}}{\\textup{sup}}|f_{k}(\\varphi(z))|+\\varepsilon C.\n\\end{eqnarray*}\nSince $\\{f_{k}\\}_{k\\in\\mathbb{N}}$ uniformly converges to zero on any compact subset of $\\mathbb{D}$, there exists a $ K\\in\\mathbb{Z}^{+}$ such that if $k>K,$  we have  $ \\underset {|\\varphi(z)|\\leq r_{0}}{\\textup{sup}}|f_{k}(\\varphi(z))|<\\varepsilon. $\nTherefore, $\\|V_{\\varphi}^{g}f_{k}\\|_{B_{\\psi}}\\rightarrow0$ as $k\\rightarrow\\infty$. By Lemma 2.4, we see that $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is compact.\n\nConversely, suppose that $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is compact, then it is clear that $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is bounded. Let $\\{z_{k}\\}_{k\\in\\mathbb{N}}$ be sequence in $\\mathbb{D}$ such that $\\underset {k\\rightarrow\\infty}{\\textup{lim}}|\\varphi(z_{k})|=1$. Let $$f_{k}(z)=\\frac{K(z,\\varphi(z_{k}))}{\\sqrt{K(\\varphi(z_{k}),\\varphi(z_{k}))}},$$\nthen, $f_{k}\\in AL_{\\psi}^{2}(\\mathbb{D})$ and $\\|f_{k}\\|_{L_{\\psi}^{2}}=1$.\n\nSince the set consisting of polynomials is dense in $ AL_{\\psi}^{2}(\\mathbb{D})$, for any $\\varepsilon>0$ and $f\\in AL_{\\psi}^{2}(\\mathbb{D})$,  there exists a polynomial $P_{f,\\varepsilon}(z)\\in AL_{\\psi}^{2}(\\mathbb{D})$  such that\n$$\\|P_{f,\\varepsilon}-f\\|_{L_{\\psi}^{2}}<\\frac{\\varepsilon}{2}.$$\n\nAs\n\\begin{eqnarray*}\n|\\langle f_{k},f\\rangle|&\\leq& |\\langle f_{k},f-P_{f,\\varepsilon}\\rangle|+|\\langle P_{f,\\varepsilon},f_{k}\\rangle|\\\\\n&\\leq&\\|f_{k}\\|_{L_{\\psi}^{2}}\\|f-P_{f,\\varepsilon}\\|_{L_{\\psi}^{2}}+|\\langle P_{f,\\varepsilon},f_{k}\\rangle|\n\\end{eqnarray*}\nby Lemma 2.2 and Definition 1.1, we have $$\\sqrt{K(z,z)}\\geq C_{1}\\tau(z)^{-1}e^{\\psi(z)}\\geq\\frac{C_{2}e^{\\psi(z)}}{1-|z|}$$ for some positive constants $C_{1}$ and $C_{2}$.\nNotice that $\\lim_{k\\rightarrow \\infty}|\\varphi(z_k)|=0$, we have\n$$\\langle P_{f,\\varepsilon},f_{k}\\rangle=\\langle P_{f,\\varepsilon},\\frac{K_{\\varphi(z_{k})}}{\\|K_{\\varphi(z_{k})}\\|_{L_{\\psi}^{2}}}\\rangle\n=\\frac{1}{\\|K_{\\varphi(z_{k})}\\|_{L_{\\psi}^{2}}}P_{f,\\varepsilon}(\\varphi(z_{k}))\\rightarrow 0~~~~\n(k\\rightarrow\\infty)$$\nThat means that $f_{k}$ weakly converges to zero  as $k\\rightarrow\\infty$.\\\\\n\nBecause $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is compact, we see that $\\|V_{\\varphi}^{g}f_{k}\\|_{B_{\\psi}}\\rightarrow 0 ~~~~(k\\rightarrow\\infty)$. From the following fact\n\\begin{eqnarray*}\n\\|V_{\\varphi}^{g}f_{k}\\|_{B_{\\psi}}&\\geq&\\underset {z\\in\\mathbb{D}}{\\textup{sup}}|f_{k}(\\varphi(z))||g'(z)|e^{-2\\psi(z)}\\\\\n&\\geq&|f_{k}(\\varphi(z_{k}))||g'(z_{k})|e^{-2\\psi(z_{k})}\\\\\n&=&\\sqrt{K(\\varphi(z_{k}),\\varphi(z_{k}))}|g'(z_{k})|e^{-2\\psi(z_{k})}\n\\end{eqnarray*}\nwe immediately obtain that $\\underset {|\\varphi(z)|\\rightarrow1}{\\textup{lim}}\\sqrt{K(\\varphi(z),\\varphi(z))}|g'(z)|e^{-2\\psi(z)}=0$.\nThe proof is completed.\n\\end{proof}\n\n\n\n\\begin{thm}\nLet $\\varphi$ be an analytic self-map of the unit disk $\\mathbb{D}$ and $g\\in H(\\mathbb{D})$, $\\psi\\in\\mathcal{D}$. Then the operator $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is compact if and only if $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is bounded and $$\\lim_{|z|\\rightarrow 1}e^{\\psi(\\varphi(z))-2\\psi(z)}\\sqrt{\\Delta\\psi(\\varphi(z))}|g'(z)|=0.$$\n\\end{thm}\n\\begin{proof}\nAt first, we note that $$\\lim_{|z|\\rightarrow 1}e^{\\psi(\\varphi(z))-2\\psi(z)}\\sqrt{\\Delta\\psi(\\varphi(z))}|g'(z)|=0$$ equals to\n$$\\underset {|z|\\rightarrow1}{\\textup{lim}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|=0.$$\n\nFirstly, we prove the sufficiency . Let $K=\\{f:f\\in AL_{\\psi}^{2}(\\mathbb{D}), \\|f\\|_{L_{\\psi}^{2}}\\leq1\\}$. As $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is bounded, $\\{V_{\\varphi}^{g}f:f\\in K\\}$ is the bounded closed set of $B_{\\psi,0}(\\mathbb{D})$. It suffices to show that $\\{V_{\\varphi}^{g}f:f\\in K\\}$ is compact in $B_{\\psi,0}(\\mathbb{D})$. By Lemma 2.5, it is only to prove\n$$\\underset {|z|\\rightarrow1}{\\textup{lim}}\\underset {\\|f\\|_{L_{\\psi}^{2}}\\leq1}{\\textup{sup}}e^{-2\\psi(z)}|(V_{\\varphi}^{g}f)^{'}(z)|=0.$$\nBy lemma 2.3, for any $ f\\in K,$ we have\n\\begin{eqnarray*}\ne^{-2\\psi(z)}|(V_{\\varphi}^{g}f)^{'}(z)|&=&e^{-2\\psi(z)}|f(\\varphi(z))||g^{'}(z)|\\\\\n&\\leq&\\|f\\|_{L_{\\psi}^{2}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|\\\\\n&\\leq&\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|.\n\\end{eqnarray*}\nNote that the condition  $\\underset {|z|\\rightarrow1}{\\textup{lim}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|=0$, we have\n$$\\underset {|z|\\rightarrow1}{\\textup{lim}}\\underset {\\|f\\|_{L_{\\psi}^{2}}\\leq1}{\\textup{sup}}e^{-2\\psi(z)}|(V_{\\varphi}^{g}f)^{'}(z)|=0.$$\nTherefore, the operator $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is compact.\n\nSecondly, we will prove the necessity. Suppose $V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is compact, it is obvious that\n$V_{\\varphi}^{g}:AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi}(\\mathbb{D})$ is compact. By Theorem 3.3, we have\n$$\\underset {|\\varphi(z)|\\rightarrow1}{\\textup{lim}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|=0.$$\nThat is, for any $\\varepsilon>0$, there exists an $r\\in(0,1)$, such that $$\\qquad\\qquad\\qquad\\qquad\\qquad\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|<\\varepsilon.\\quad\\quad\\qquad\\qquad\\qquad(*)$$\nwhen $r<\\varphi(z)<1$.\\\\\n\nSince $V_{\\varphi}^{g}: AL_{\\psi}^{2}(\\mathbb{D})\\rightarrow B_{\\psi,0}(\\mathbb{D})$ is bounded,\nby Theorem 3.2, we have\n$$\\underset {|z|\\rightarrow1}{\\textup{lim}}e^{-2\\psi(z)}|g'(z)|=0.$$\nLet $\\varepsilon^{'}=\\frac{\\varepsilon}{C_{r}}$, there exists a positive real number $\\sigma>0$, such that $$e^{-2\\psi(z)}|g'(z)|<\\frac{\\varepsilon}{C_{r}}$$\nwhen $\\sigma<|z|<1$, where $C_{r}$ is the upper bound of $\\sqrt{K(\\varphi(z),\\varphi(z))}$ when $|\\varphi(z)|\\leq r$.\nTherefore, we have $$\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|\n<C_{r}\\frac{\\varepsilon}{C_{r}}=\\varepsilon.$$\nwhen $\\sigma<|z|<1$ and $|\\varphi(z)|\\leq r$.\\\\\n\\par\nCombines with $(\\ast)$, we see that $\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|<\\varepsilon$ when $\\sigma<|z|<1$.\nTherefore, $$\\underset {|z|\\rightarrow1}{\\textup{lim}}\\sqrt{K(\\varphi(z),\\varphi(z))}e^{-2\\psi(z)}|g'(z)|=0.$$\nThe proof is completed.\n\\end{proof}\n\n\n\n\n\n\\end{section}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction}\nThe ensemble Kalman filter \\cite{evensen1994sequential,burgers1998analysis,evensen2009data}, one of the most widely applied data assimilation  algorithms \\cite{asch2016data,law2015data,reich2015probabilistic}, uses a Monte Carlo approach to provide a non-linear approximation to the Kalman filter~\\cite{kalman1960new}. In the typical case of an undersampled ensemble the algorithm requires correction procedures such as inflation~\\cite{anderson2001ensemble}, localization~\\cite{hunt2007efficient, petrie2008localization, anderson2012localization,Sandu_2015_SCALA,Sandu_2017_Covariance-Cholesky,zhang2010ensemble}, and ensemble subspace enrichment~\\cite{Sandu_2015_covarianceShrinkage, Sandu_2019_Covariance-parallel,Sandu_2014_EnKF_SMF}. \n\nHybrid data assimilation \\cite{hamill2000hybrid} is typically an umbrella term for assimilation techniques that combine both offline-estimated climatological covariances with their online-estimated statistical counterparts. These methods are often thought of as heuristic corrections, but in fact stem from  statistically rigorous covariance shrinkage techniques.\n\nThis work is based on enriching the ensemble subspace through the use of climatological covariances. Previous work~\\cite{Sandu_2015_covarianceShrinkage, Sandu_2019_Covariance-parallel} proposed augmenting the covariance estimates derived from the ensemble by a full rank shrinkage covariance matrix approximation. In this work we consider augmenting the physical ensemble with synthetic members drawn from a normal distribution with a possibly low rank covariance matrix derived from \\textit{a priori} information such a climatological information or method of snapshots. We show that this is equivalent to a stochastic implementation of the shrinkage covariance matrix estimate proposed in ~\\cite{Sandu_2015_covarianceShrinkage, Sandu_2019_Covariance-parallel}, and therefore augmenting the physical ensemble with synthetic members enriches the rank of the covariance matrix, and nudges the resulting covariance estimate toward the true covariance.\n\n\\section{Background}\nOur aim is to understand the behavior of an evolving natural phenomenon. The evolution of the natural phenomenon is approximated by an imperfect dynamical model\n\\begin{equation}\n    X_i = \\!M_{(i-1)\\to i}(X_{i-1}) + \\Xi_i,\n\\end{equation}\nwhere $X_{i-1}$ is a random variable (RV) whose distribution represents our uncertainty in the state of the system at time $i-1$, $\\!M_{(i-1)\\to i}$ is the (imperfect) dynamical model, $\\Xi_i$ is a RV whose distribution represents our uncertainty in the additive modeling error, and $X_i$ is the RV whose distribution represents our uncertainty in the (forecasted) state at time $i$.\n\nOne collects noisy observations of the truth:\n\\begin{equation}\n    \\*y^\\|o_i = \\!H_i(\\*x_i^\\|t) + \\*\\eta_i,\n\\end{equation}\nwhere $\\*x^\\|t$ represents the true state of nature represented in model space, $\\!H_i$ is the (potentially non-linear) observation operator, $\\*\\eta_i$ is a RV whose distribution represents our uncertainty in the observations, and $\\*y^\\|o_i$ are the observation values, assumed to be realizations of an observation RV $Y_i$. Take $n$ to be the dimension of the state-space, and $m$ to be the dimension of the observation space.\n\nThe goal of data assimilation is to find the \\textit{a posteriori} estimate of the state given the observations, which is typically achieved through Bayes' theorem. At time $i$ we have:\n\\begin{equation}\n    \\pi(X_i|Y_i) \\propto \\pi(Y_i|X_i)\\,\\pi(X_i).\n\\end{equation}\nIn typical Kalman filtering the assumption of Gaussianity is made, whereby the states at all times, as well as the additive model and observation errors, are assumed to be Gaussian and independently distributed. Specifically one assumes $\\Xi_i \\sim  \\!N(\\*0, \\*Q_i)$ and $\\*\\eta_i \\sim \\!N(\\*0,\\mathbf{R}_i)$. \n\nIn what follows we use the following notation. The \\textit{a priori} estimates at all times are represented with the superscript $\\square^\\|f$, for forecast (as from the imperfect model), and the \\textit{a posteriori} estimates are represented with the superscript $\\square^\\|a$, for analysis (through a DA algorithm).\n\n\n\\subsection{Ensemble Transform Kalman Filter}\nForecasting with an ensemble of coarse models has proven to be a more robust methodology than forecasting with a single fine model~\\cite{kalnay2003atmospheric}. Ensemble Kalman filtering aims to utilize the ensemble of forecasted states to construct empirical moments and use them to implement the Kalman filter formula. The Ensemble Transform Kalman Filter (ETKF) computes an optimal transform of the prior ensemble member states to the posterior member states; for Gaussian distributions the optimal transform is described by a symmetric transform matrix.\n\nWe now describe the standard ETKF. Let $\\*X^\\|a_{i-1} = [\\*x^{(1),\\|a}_{i-1},\\dots \\*x^{(N),\\|a}_{i-1}]$ represent the $N$--members analysis ensemble at time $i-1$. The forecast step is:\n\\begin{equation}\n    \\*x_i^{(k),\\|f} = \\!M_{(i-1)\\to i}(\\*x_{i-1}^{(k),\\|a}) + \\*\\xi_i^{(k)},\\quad k=1,\\dots,N,\n\\end{equation}\nwhere $\\*\\xi_i^{(k)}$ is a random draw from $ \\!N(\\*0,\\*Q_i)$. \n\nThe ETKF analysis step reads:\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{eq:ETKF-analysis-anomalies}\n    \\*A^\\|a_i &=& \\*A^\\|f_i\\, \\*T_i, \\\\\n\\label{eq:ETKF-analysis-mean}\n    \\bar{\\*x}^\\|a &=& \\bar{\\*x}^\\|f + \\*A^\\|a_i\\, \\*Z^{\\|a,\\mathsf{T}}\\,\\mathbf{R}^{-1}\\,\\*d_i,\n\\end{eqnarray}\n\\end{subequations}\nwhere\n\\begin{subequations}\n\\begin{align}\n\\label{eq:transform-matrix}\n\\*T_i &= {\\left(\\mathbf{I} - \\*Z_i^{\\|f,\\mathsf{T}}\\,\\S_i^{-1}\\,\\*Z_i^\\|f\\right)}^{\\frac{1}{2}}, \\\\\n\\label{eq:S-matrix}\n\\S_i &= \\*Z_i^\\|f\\,\\*Z_i^{\\|f,\\mathsf{T}} + \\mathbf{R}_i, \\\\\n\\*A^\\|f_i &= \\frac{1}{\\sqrt{N-1}}\\left(\\*X^\\|f_i - \\overline{\\*x}^\\|f_i\\,\\*1^\\mathsf{T}\\right), \\\\\n\\*Z^\\|f_i &= \\frac{1}{\\sqrt{N-1}}\\left(\\!H(\\*X^\\|f) - \\overline{\\!H(\\*X^\\|f)}\\, \\*1^\\mathsf{T}\\right), \\\\\n\\*d_i &= \\*y^\\|o_i - \\overline{\\!H(\\*X^\\|f_i)},\\\\\n\\overline{\\*x}^\\|f_i &= \\frac{1}{N}\\sum_{k=1}^N \\*X^{\\|f,(k)}_i,\\\\\n\\overline{\\!H(\\*X^\\|f_i)} &= \\frac{1}{N}\\sum_{k=1}^N \\!H(\\*X^{\\|f,(k)}_i).\n\\end{align}\n\\end{subequations}\nHere the unique symmetric square root of the matrix is used, as there is evidence of that option being the most numerically stable~\\cite{sakov2008implications}. Also, it is common practice to approximate $\\*Z^\\|a$ by\n\\begin{equation}\n\\label{eq:ZaAssum}\n    \\*Z^\\|a \\approx \\*Z^\\|f\\,\\*T_i,\n\\end{equation}\nalthough in reality $\\*Z^\\|a$ is implicitly defined from the analysis ensemble $\\*X^\\|a$ as follows:\n\\begin{equation}\n\\label{eq:Za}\n\\begin{split}\n    \\*X^\\|a_i &= \\bar{\\*x}^\\|a_i\\,\\*1^\\mathsf{T} + \\sqrt{N-1}\\*A^\\|a_i, \\\\\n    \\*Z^\\|a &= \\frac{1}{\\sqrt{N-1}}\\left(\\!H(\\*X^\\|a) - \\overline{\\!H(\\*X^\\|a)}\\,\\*1^\\mathsf{T}\\right).\n\\end{split}    \n\\end{equation}\nNote that equation~\\eqref{eq:Za} is automatically satisfied by~\\eqref{eq:ZaAssum} when $\\!H$ is a linear operator. Additionally, this implicitly defines a fixed point iteration which should converge when the linear operator is sufficiently smooth. In this paper we make the assumption that equation~\\eqref{eq:ZaAssum} is exact.\n\nThe empirical forecast covariance estimate \n\\begin{equation}\n\\label{eq:empirical-forecast-covariance}\n\\Ct{X^\\|f_i}{X^\\|f_i} = \\*A^\\|f_i\\,\\*A^{\\|f,\\mathsf{T}}_i\n\\end{equation}\nis not perfect.  In order to improve the performance of EnKF, inflation is applied to the ensemble anomalies,\n\\begin{equation}\n    \\*A^\\|f_i \\leftarrow \\alpha \\, \\*A^\\|f_i,\n\\end{equation}\nbefore any other computation is performed (meaning that it is also applied to the observation anomalies, $\\*Z^\\|f_i$ as well). The inflation parameter $\\alpha>1$ is typically chosen in a heuristic manner.\n\n\\subsection{Covariance localization}\nTraditional state-space localization of the empirical covariance \\eqref{eq:empirical-forecast-covariance} is done by tapering, i.e., by using a Schur product\n\\begin{equation}\n \\*B^\\|f_i =  \\*\\rho_i \\circ \\Ct{X^\\|f_i}{X^\\|f_i},\n\\end{equation}\nwith the localization matrix $\\*\\rho$ contains entries that are progressively smaller as the distance between the corresponding variables increases.\nTapering methods for localization can also be thought of as a type of shrinkage \\cite{chen2012shrinkage}.\n\nThe localized ETKF (LETKF)~\\cite{hunt2007efficient} is a localization approach to the ETKF. The LETKF and its variants are considered to be one of the state-of-the-art EnKF methods.\nIn the state-space approach to the LETKF, each state space variable $[u]_j$ is assimilated independently of all others, with the observation space error covariance inverse replaced by\n\\begin{equation}\\label{eq:rloc}\n    \\*R_i^{-1} \\xleftarrow{} \\*\\rho_{i, j} \\circ \\*R_i^{-1},\n\\end{equation}\nwith the matrix $\\*\\rho_{i,[j]}$ being a diagonal matrix representing the decorrelation factor between all observation space variables from the $j$th state space variable.  Each diagonal element represents the tapering factor, and is often chosen to be some function of distance from the state-space variable being assimilated to the corresponding observation-space variable. The implicit assumption is that all observations are independent of each other, in both the observation error ($\\*R_i$ is diagonal), and forecast error ($\\*Z^\\|f\\*Z^{\\|f,\\mathsf{T}}$ is assumed to be diagonal).\n\n\n\n\n\\subsection{Covariance shrinkage}\nIn statistical literature\\cite{chen2009shrinkage,chen2010shrinkage,chen2011robust,ledoit2004well} covariance shrinkage refers to describe the methodology under which a statistical covariance derived from a set of samples is made to approach the ``true'' covariance from which the set of samples is derived. For the vast majority of statistical applications, there is no additional apriori knowledge about the distribution of the samples, thus assumptions such as Gaussianity and sphericity are made. In data assimilation applications, however, climatological estimates of covariance are extremely commonplace.\n\nAssume that there exists a target covariance matrix $\\!P$ that represents the \\textit{a priori} knowledge about the error spatial correlations.\nIn this paper we focus on an additive shrinkage covariance structure which is a linear combination of the empirical covariance \\eqref{eq:empirical-forecast-covariance} and the target covariance,\n\\begin{equation}\n\\label{eq:shrinkage-covariance}\n    \\*B^\\|f_i = \\gamma_i\\,\\mu_i\\,\\!P + (1-\\gamma_i)\\,\\Ct{X^\\|f_i}{X^\\|f_i},\n\\end{equation}\nwith $\\gamma_i$ representing the shrinkage factor and $\\mu_i$ representing a scaling factor. By employing a general target matrix $\\!P$, a closed-form expression to compute the weight value $\\gamma_i$ is proposed in  \\cite{Stoica2008,Zhu2011}. In this derivation, weights are computed as follows:\n\\begin{equation}\n    \\gamma_i=\\min\\left(\\frac{\\frac{1}{N^2} \\cdot\\sum_{k=1}^N\\norm{\\*x_i^{(k),\\|f}-\\overline{\\*x^\\|f_i}}^4 -\\frac{1}{N}\\cdot \\norm{ \\Ct{X^\\|f_i}{X^\\|f_i}}^2}{\\norm{\\Ct{X^\\|f_i}{X^\\|f_i}-\\!P}^2},1\\right), \\label{eq:KA}\n\\end{equation}\nbut as such an estimate is expensive to compute in an operational setting well will settle for a more computationally inexpensive method.\nNo assumptions about the structure of $\\!P$ are made to compute $\\gamma_i$. The general form \\eqref{eq:shrinkage-covariance} can be reduced to a standard form where the target matrix is the (scaled) identity by defining the statistical covariance $\\!C_i = \\!P^{-\\frac{1}{2}}\\Ct{X^\\|f_i}{X^\\|f_i}\\!P^{-\\frac{1}{2}}$. The corresponding scaling factor, $\\mu_i$ is then the average covariance of $\\!C_i$, \n\\begin{equation}\n\\label{eq:mu}\n    \\mu_i = \\frac{\\tr(\\!C_i)}{n},\n\\end{equation}\nwith the new target $\\mu_i\\*I$ representing a spherical climatological assumption on $\\!C_i$.\n\nThe choice of $\\gamma_i$ is extremely important. In this paper we focus on the Rao-Blackwellized Ledoit-Wolf (RBLW) estimator~\\cite{chen2009shrinkage} (equation~(9) in~\\cite{Sandu_2017_Covariance-Cholesky}),\n\\begin{equation}\n\\label{eq:RBLW}\n\\gamma_{i,\\text{RBLW}} = \\min \\left[\\vcenter{\\hbox{$\\displaystyle\\frac{N - 2}{N(N+2)} + \\frac{(n + 1)N - 2}{\\hat{U}_iN(N+2)(n-1)}$}},\\,\\,\\, 1\\right],\n\\end{equation}\nwhere the sphericity factor\n\\begin{equation}\n\\label{eqn:sphericity}\n    \\hat{U}_i = \\frac{1}{n - 1}\\left(\\frac{n \\tr(\\!C_i^2)}{\\tr^2(\\!C_i)} - 1\\right)\n\\end{equation}\nmeasures how similar the correlation structures of the sample covariance and target covariance are. Note that if our samples are also used to calculate the sample mean, the effective sample size of the sample covariance is smaller by one, therefore for most practical applications one replaces $N$ by $N-1$ in \\eqref{eq:RBLW}.\n\nThe RBLW estimator is valid under Gaussian assumptions about the samples from which the sample covariance matrix is derived, however is only considered to be accurate for an oversampled sample covariance matrix.\n\n\nThere are two major issues with the application of the shrinkage estimator in the EnKF, both related to its reliance on the sphericity of $\\!C_i$. First, when operating in the undersampled regime $N \\ll n$, the estimate $\\!C_i$ is also undersampled, and the problem of ``spurious correlations'' will affect the measure of sphericity \\eqref{eqn:sphericity}. The second related issue regards the climatological estimate $\\!P$. Unless the climatological estimate accurately measures the correlation structure of the sample covariance, the shrinkage estimate could potentially be very poor. The long-term accuracy of the climatological estimate to the covariance is thus of great importance. \n\nNote that $\\!P$ is not required to be invertible. Commonly, a reduced spectral version of $\\!P$ is known, $\\!P = \\!U\\,\\!L\\,\\!U^*$, with the $\\!L$ being a diagonal matrix of $r\\ll n$ spectral coefficients, and $\\!U$ being an $n\\times r$ matrix of orthonormal coefficients. The canonical symmetric pseudo-inverse square-root of $\\!P$ would therefore be $\\!P^{-\\frac{1}{2}} = \\!U\\,\\!L^{-\\frac{1}{2}}\\,\\!U^*$. If $\\sigma_i$ is the $i$-th singular value of $\\!P^{-1\/2}\\*A^\\|f$, then the traces appearing in \\eqref{eqn:sphericity} can be computed as:\n\\begin{gather*}\n    \\tr\\left(\\!C\\right) = \\sum_{i=1}^{N-1}\\sigma_i^2,\\qquad\n    \\tr\\left(\\!C^2\\right) = \\sum_{i=1}^{N-1}\\sigma_i^4.\n\\end{gather*}\n\n\n\\section{ETKF implementation with stochastic shrinkage covariance estimates}\n\n\n\nEnsemble methods propagate our uncertainty about the dynamics of the system. Application of Bayes' rule requires that all available information is used~\\cite{jaynes2003probability}. Therefore, we attempt to use the locally known climatological information in conjunction with our ensemble information.\nIf the dynamical system is locally (in time) stationary, climatologies about the local time roughly describe a measure of averaged-in-space uncertainty. \n\nNino-Ruiz and Sandu ~\\cite{Sandu_2015_covarianceShrinkage, Sandu_2019_Covariance-parallel} proposed to replace the empirical covariance in EnKF with a shrinkage covariance estimator~\\eqref{eq:shrinkage-covariance}. They showed that this considerably improves the analysis at a modest additional computational cost. Additional, synthetic ensemble members drawn from a normal distribution with covariance $\\*B^\\|f$ are used to decrease the sampling errors.\n\nIn this work we develop an implementation of ETKF with a  stochastic shrinkage covariance estimator~\\eqref{eq:shrinkage-covariance}. Rather than computing the covariance estimate ~\\eqref{eq:shrinkage-covariance}, we build a synthetic ensemble by sampling directly from a distribution with covariance $\\mu_i \\!P$.\nThe anomalies of this synthetic ensemble are independent of the anomalies of the forecast EnKF ensemble. \n\nTo be specific, let $\\!X^\\|f \\in \\Re^{n \\times M}$ be a synthetic ensemble of $M$ members (as opposed to the dynamic ensemble $\\*X^\\|f_i$ with $N$ members) drawn from a climatological probability density. We denote the variables related to the synthetic ensemble by calligraphic letters. \n\nA major concern is the choice of distribution. As sampling from the dynamical manifold is impractical,  heuristic assumptions are made about the distributions involved. A useful known heuristic is the principle of maximum entropy (PME)~\\cite{jaynes2003probability}. Given the PME, and the assumption that we know both the mean and covariance of our assumed distribution, and the assumption that the dynamics are supported over all of $\\Re^n$, there is one standard interpretation of the information contained in the synthetic ensemble $\\!X^\\|f_i$. Assuming that the mean and covariance are known information (through sampling),  a Gaussian assumption on the synthetic ensemble is warranted,\n\\begin{equation}\n\\!X^\\|f_i \\sim \\mathcal{N}(\\bar{\\*x}^\\|f_i,\\mu_i\\,\\!P).\n\\end{equation}\nAlternatively, not explored in this paper, a Laplace assumption on the synthetic ensemble can be assumed. The Laplace assumption has obvious parallels to robust statistics methods in data assimilation~\\cite{rao2017robust}. By sampling from a Laplace distribution, outlier behaviour can more readily be captured with fewer samples. However, the concentration of samples around the mean is also increased, thereby requiring more samples to better represent the covariance. The Gaussian assumption appears more natural in Kalman filter-based methods, and thus the rest of the paper will make this assumption.\n\n\nThe synthetic ensemble anomalies in the state and observation spaces are:\n\\begin{equation}\n\\label{eq:synthetic-anomalies}\n\\begin{split}\n    \\!A^\\|f_i &= \\frac{1}{\\sqrt{ M-1}}\\left(\\!X^\\|f_i - \\overline{\\!X}^\\|f_i\\,\\*1^\\mathsf{T}\\right) \\in \\Re^{n\\times M},\\\\\n    \\!Z^\\|f_i &= \\frac{1}{\\sqrt{ M-1}}\\left(\\!H(\\!X^\\|f_i) - \\overline{\\!H(\\!X^\\|f_i)}\\,\\*1^\\mathsf{T}\\right) \\in \\Re^{m\\times M}.\n\\end{split}\n\\end{equation}\n\nThe shrinkage  estimator \\eqref{eq:shrinkage-covariance} of the forecast error covariance for $\\*B^\\|f_i$ is represented in terms of synthetic and forecast anomalies as:\n\\begin{equation}\n\\label{eq:Bf-shrinkage-stochastic}\n\\widetilde{\\*B}^\\|f_i = \\gamma_i\\,\\!A^\\|f_i\\,\\!A^{\\|f,\\mathsf{T}}_i + (1-\\gamma_i)\\,\\*A^\\|f_i\\,\\*A^{\\|f,\\mathsf{T}}_i.\n\\end{equation}\nThe Kalman filter formulation ~\\cite{kalman1960new} yields the following analysis covariance matrix:\n\\begin{equation}\n\\label{eq:Ba-shrinkage-stochastic}\n\\widetilde{\\*B}^\\|a_i = \\widetilde{\\*B}^\\|f_i - \\widetilde{\\*B}^\\|f_i\\,\\*H_i^\\mathsf{T}\\,\\S_i^{-1}\\,\\*H_i \\,\\widetilde{\\*B}^\\|f_i,\n\\end{equation}\nwhere $\\S_i$ will be discussed later. \n\nUsing the forecast error covariance estimate  \\eqref{eq:Bf-shrinkage-stochastic} in \\eqref{eq:Ba-shrinkage-stochastic} leads to the following analysis covariance:\n\\begin{equation}\n\\begin{split}\n\\widetilde{\\*B}^\\|a_i &= \\gamma_i\\,\\!A^\\|f_i\\,\\!A^{\\|f,\\mathsf{T}}_i + (1-\\gamma_i)\\,\\*A^\\|f_i\\,\\*A^{\\|f,\\mathsf{T}}_i \\\\\n&\\quad - \\left(\\gamma_i\\,\\!A^\\|f_i\\,\\!Z^{\\|f,\\mathsf{T}}_i + (1-\\gamma_i)\\,\\*A^\\|f_i\\,\\*Z^{\\|f,\\mathsf{T}}_i\\right)\\,\\S^{-1}_i\\,\\left(\\gamma_i\\,\\!Z^\\|f_i\\,\\!A^{\\|f,\\mathsf{T}}_i + (1-\\gamma_i)\\,\\*Z^\\|f_i\\,\\*A^{\\|f,\\mathsf{T}}_i\\right),\n\\end{split}\n\\label{eq:goal-cov}\\tag{goal-cov}\n\\end{equation}\nwhich we refer to as the~ ``target'' analysis covariance formula.\n\nThe goal of our modified ensemble Kalman filter is to construct an $N$-member analysis ensemble such that the anomalies $\\*A^\\|a_i$ \\eqref{eq:ETKF-analysis-anomalies} represent the~\\eqref{eq:goal-cov} analysis covariance as well as possible:\n\\begin{equation}\n\\label{eq:goal-an}\\tag{goal-an}\n\\textnormal{Find}~~\\*A^\\|a_i \\in \\Re^{n \\times N}~~\\textnormal{such that}:\\quad\n \\*A^\\|a_i \\,\\*A^{\\|a,\\mathsf{T}}_i \\approx \\widetilde{\\*B}^\\|a_i.\n\\end{equation}\nIn the proposed method, we enrich our forecast ensemble in a way that closely approximates the shrinkage covariance~\\eqref{eq:shrinkage-covariance}.\nAn alternative approach, where we transform the physical ensemble and the surrogate ensemble as separate entities that are only related by the common information in their transformations, is discussed in Appendix~\\ref{sec:appendix}.\n\n\n\\subsection{The method}\nWe enrich the ensembles of forecast anomalies with synthetic anomalies \\eqref{eq:synthetic-anomalies}:\n\\begin{equation}\n\\label{eq:type1-enriched-ensembles}\n\\begin{split}\n    \\#A^\\|f_i &= \\begin{bmatrix}\\sqrt{1-\\gamma_i}\\,\\*A^\\|f_i & \\sqrt{\\gamma_i}\\,\\!A^\\|f_i\\end{bmatrix} \\in \\Re^{n\\times (N + M)},\\\\\n    \\#Z^\\|f_i &= \\begin{bmatrix}\\sqrt{1-\\gamma_i}\\,\\*Z^\\|f_i & \\sqrt{\\gamma_i}\\,\\!Z^\\|f_i\\end{bmatrix} \\in \\Re^{m\\times (N + M)}.\n\\end{split}\n\\end{equation}\nNext, we define a transform matrix \\eqref{eq:ETKF-analysis-anomalies} that is applied to the enriched ensemble \\eqref{eq:type1-enriched-ensembles}, and leads to an analysis ensemble that represents the target analysis covariance \\eqref{eq:goal-an}. Specifically, we search for a transform matrix $\\#T_i$ such that:\n\\begin{equation}\n\\label{eq:T1eq}\n    \\widetilde{\\*B}^\\|a_i = \\#A^\\|f_i\\,\\#T_i\\,\\#T_i^\\mathsf{T}\\,\\#A^{\\|f,\\mathsf{T}}_i.\n\\end{equation}\nUsing the extended ensembles \\eqref{eq:T1eq} the~\\eqref{eq:goal-cov} becomes\n\\begin{equation}\n\\label{eq:t1ba}\n    \\widetilde{\\*B}^\\|a_i = \\#A^\\|f_i\\,\\left(\\mathbf{I}_{(N+M) \\times (N+M)} - \\#Z^{\\|f,\\mathsf{T}}_i\\,\\S^{-1}_i\\,\\#Z^\\|f_i\\right)\\,\\#A^{\\|f,\\mathsf{T}}_i,\n\\end{equation}\nwhere, from \\eqref{eq:S-matrix},\n\\begin{equation}\n    \\S_i = \\#Z^\\|f_i\\,\\#Z^{\\|f,\\mathsf{T}}_i + \\mathbf{R}_i.\n\\end{equation}\nThe transform matrix \\eqref{eq:transform-matrix} is a square root of ~\\eqref{eq:T1eq}:\n\\begin{equation}\n\\label{eq:T-symmetric-sqrt}\n    \\#T_i = {\\left(\\mathbf{I}_{(N+M) \\times (N+M)} - \\#Z^{\\|f,\\mathsf{T}}_i\\,\\S^{-1}_i\\,\\#Z^\\|f_i\\right)}^{\\frac{1}{2}}.\n\\end{equation}\nWe compute the analysis mean using the shrinkage covariance estimate. From  \\eqref{eq:ETKF-analysis-mean}:\n\\begin{align}\n\\label{eq:type1-mean}\n    \\bar{\\*x}^\\|a_i &= \\bar{\\*x}^\\|f_i + \\#A^\\|f_i\\,\\#T_i\\,\\#T_i^\\mathsf{T}\\,\\#Z^{\\|f,\\mathsf{T}}_i\\,\\mathbf{R}^{-1}_i\\,\\*d_i,\n\\end{align}\nwhere the full analysis covariance  estimate ~\\eqref{eq:t1ba} is used.\nIn addition, we achieve the \\eqref{eq:goal-an} by keeping the first $N$ members of the transformed extended ensemble, or equivalently, the first $N$ columns of the symmetric square root \\eqref{eq:T-symmetric-sqrt}.  From \\eqref{eq:ETKF-analysis-anomalies} we have:\n\\begin{align}\n\\label{eq:type1-anomalies}\n    \\*A^\\|a_i &= \\frac{1}{\\sqrt{1-\\gamma_i}}\\,\\bigl[\\#A^\\|f_i\\,\\#T_i\\bigr]_{:,1:N} = \\frac{1}{\\sqrt{1-\\gamma_i}} \\,\\#A^\\|f_i\\,\\breve{\\#T}_i,\n    \\quad \\breve{\\#T}_i = \\left[\\#T_i\\right]_{:,1:N}.\n\\end{align}\nAn alternative approach to achieve the \\eqref{eq:goal-an} is to look for a low-rank, approximate square root instead of the symmetric square root \\eqref{eq:T-symmetric-sqrt}. Specifically, we seek a transformation matrix $\\widehat{\\#T}_i$ such that:\n\\begin{equation}\n\\label{eq:T-lowrank-sqrt}\n\\widehat{\\#T}_i \\in  \\Re^{(N+M) \\times N}, \\qquad\n    \\widehat{\\#T}_i\\,\\widehat{\\#T}_i^T \\approx \\mathbf{I}_{(N+M) \\times (N+M)} - \\#Z^{\\|f,\\mathsf{T}}_i\\S^{-1}_i\\#Z^\\|f_i.\n\\end{equation}\nThe calculation of the symmetric square root \\eqref{eq:T-symmetric-sqrt} requires an SVD of the right hand side matrix. With the same computational effort one can compute the low rank transformation:\n\\begin{equation}\n\\label{eq:T-symmetric-sqrt2}\n\\begin{split}\n\\mathbf{U}\\, \\boldsymbol{\\Sigma} \\, \\mathbf{U}^T &= {\\left(\\mathbf{I} - \\#Z^{\\|f,\\mathsf{T}}_i\\S^{-1}_i\\#Z^\\|f_i\\right)}, \\qquad \n\\mathbf{U}, \\boldsymbol{\\Sigma} \\in  \\Re^{(N+M) \\times (N+M)};\\\\\n    \\breve{\\#T}_i &= \\mathbf{U}\\, \\boldsymbol{\\Sigma}^{1\/2} \\, [\\mathbf{U}_{1:N,:}]^T \\in  \\Re^{(N+M) \\times N} \\qquad\\textnormal{(symmetric square root \\eqref{eq:T-symmetric-sqrt})}; \\\\\n    \\widehat{\\#T}_i &= \\mathbf{U}_{:,1:N}\\, \\boldsymbol{\\Sigma}^{1\/2}_{1:N,1:N} \\in  \\Re^{(N+M) \\times N} \\qquad \\textnormal{(low rank square root \\eqref{eq:T-lowrank-sqrt})}.\n\\end{split}\n\\end{equation}\nThe mean calculation \\eqref{eq:type1-mean} is the same. The ensemble transform produces $N$ transformed ensemble members that contain ``mixed'' information from both the physical and the synthetic ensembles:\n\\begin{align*}\n\\label{eq:type2-anomalies}\n    \\*A^\\|a_i &= \\frac{1}{\\sqrt{1-\\gamma_i}} \\,\\#A^\\|f_i\\,\\widehat{\\#T}_i.\n\\end{align*}\n\n\\begin{remark}[Classical localization]\nIt is possible to combine the proposed stochastic shrinkage approach with traditional localization. The LETKF implementation~\\cite{hunt2007efficient} computes transform matrices for subsets of variables, corresponding to localized spatial domains. In a similar vein one can combine our shrinkage algorithm with classical localization, as follows. Subsets of variables of the enriched ensembles \\eqref{eq:type1-enriched-ensembles} are used to compute local transform matrices \\eqref{eq:T-symmetric-sqrt} or \\eqref{eq:T-lowrank-sqrt}, which are then applied to transform the corresponding local subsets, i.e. to compute the corresponding rows in equations \\eqref{eq:type1-anomalies} or \\eqref{eq:type1-anomalies}, respectively.\nInsofar, the authors have not observed this to have any measurable effect on the error.\n\\end{remark}\n\n\\section{Numerical experiments}\n\n\nAll test problem implementations are available in the `ODE Test Problems' suite \\cite{otp, otpsoft}.\n\n\\subsection{The Lorenz'96 model  (L96)}\n\n\nWe first consider the 40-variable Lorenz '96 problem~\\cite{lorenz1996predictability},\n\\begin{equation}\n\\label{eq:Lorenz}\n    \\left[y\\right]'_i = - \\left[y\\right]_{i-1}\\left(\\left[y\\right]_{i-2} - \\left[y\\right]_{i+1}\\right) - \\left[y\\right]_i + F, \\quad i=1,\\dots,40, \\quad F=8.\n\\end{equation}\nAssuming \\eqref{eq:Lorenz} is ergodic (thus having a constant spatio-temporal measure of uncertainty on the manifold of the attractor), we compute the target covariance matrix $\\!P$ as the empirical covariance from $10,000$ independent ensemble members run over $225$ days in the system (where $0.05$ time units corresponds to 6 hours), with an interval of 6 hours between snapshots, This system is known to have 13 positive Lyapunov exponents, with a Kaplan-Yorke dimension of about 27.1 \\cite{popov2019bayesian}. \n\nThe time between consecutive assimilation steps is $\\Delta t = 0.05$ units, corresponding to six hours in the system. All variables are observed directly with an observation error covariance matrix of $\\mathbf{R}_i = \\mathbf{I}_{40}$. The time integration of the model is performed with RK4 the fourth order Runge-Kutta scheme RK4 \\cite{hairer1991solving} with a step size $h = \\Delta t$. The problem is run over 2200 assimilation steps. The first 200 are discarded to account for model spinup. Twenty independent model realizations are performed in order to glean statistical information thereof.\n\n\\subsection{L96 assimilation results}\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figure1.pdf}\n    \\caption{Results for the L96 problem with dynamic ensembles sizes of $N=5$ and $N=14$, inflation factor $\\alpha=1.1$, and different synthetic ensemble sizes $M$. We compute the KL divergence of the rank histogram~\\eqref{eq:rank-histogram} and the RMSE~\\eqref{eq:rmse} for the methods. Error bars show two standard deviations.}\\label{fig:KLRMSEM}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.66\\linewidth]{figure2.pdf}\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\caption{The left panel presents the RMSE of the L96 model for various values of the dynamic and synthetic ensemble sizes. The right panel presents the shrinkage factor $\\gamma$ for a synthetic ensemble size of $M=100$, with error bars showing two standard deviations.}\n    \\label{fig:l96-M-versus-N-versus-G}\n\\end{figure*}\n\n\nWe assess the quality of the analysis ensembles using a rank histogram~\\cite{hamill2001interpretation}, cumulative over 20 independent runs. For a quantitative metric we compute the KL divergence from $Q$ to $P$,\n\\begin{equation}\\label{eq:rank-histogram}\n    D_{KL}\\left(P\\middle|\\middle|Q\\right) = -\\sum_k P_i\\log\\left(\\frac{P_k}{Q_k}\\right),\n\\end{equation}\nwhere $P$ is the uniform distribution and $Q$ is our ensemble rank histogram, and $P_k$ \\& $Q_k$ are the discrete probabilites associated with each bin.\n\nAdditionally, for testing the accuracy of all our methods we compute the spatio-temporal RMSE,\n\\begin{equation}\\label{eq:rmse}\n    \\sqrt{\\sum_{i=1}^K \\sum_{j=1}^n \\left[x_i\\right]_j^2},\n\\end{equation}\nwith $K$ representing the amount of snapshots at which the analysis is computed, and $\\left[x_i\\right]_j$ is the $j$th component of the state variable at time $i$.\n\nFor the given settings of a severely undersampled ensemble ($N=5$) and mild inflation ($\\alpha = 1.1$), we compare the Gaussian sampling methodology coupled to the RBLW formulation for the shrinkage factor $\\gamma$~\\eqref{eq:RBLW}, with the optimal static $\\gamma=0.85$ shrinkage factor. For a dynamic ensemble that captures the positive error growth modes ($N=14$) will will compare the RBLW estimator with the optimal static $\\gamma=0.1$. We will compare the mean and variance of the KL divergence of the rank histogram of the variable $[y]_{17}$ from the uniform, and the statistics of the spatio-temporal RMSE.\n\nThe results are reported in Figure~\\ref{fig:KLRMSEM}. For both an undersampled and sufficient ensemble, the optimal shrinkage factor has a smaller mean error, and a smaller KL divergence with less variance (top left, top middle, bottom left, and bottom middle panels). In the undersampled case, the RBLW estimator induces more variance into the RMSE (top middle panel). For the sufficiently sampled ensemble, however, the optimal static shrinkage value induced significantly more variance into the error, with the RBLW estimator reducing the error significantly (bottom middle panel). \n\nIt is possible that a better estimator than RBLW may get the `best of both worlds' and induce low error with low variance, though this is as-of-yet out of reach. This is to be expected as the RBLW estimate is only accurate in the limit of ensemble size, and there is no theory about its accuracy in the undersampled case. In the authors' experience other estimators such as OAS, while having the theoretically desired properties, perform empirically worse in conjunction with ensemble methods. Currently, for a modest reduction in accuracy, one of the hyperparameters can be {estimated online} by the methodology.\n\nFor the second round of experiments with Lorenz '96, reported in Figure~\\ref{fig:l96-M-versus-N-versus-G}, we comparing analysis errors when the synthetic and dynamic ensemble sizes are modified (left panel). It is evident that increases in both dynamic and synthetic ensemble size lead to lower error. In the right panel we also compare dynamic ensemble size the values of $\\gamma$ that are produced. It is clear that an increase in dynamic ensemble size decreases the need for shrinkage.\n\n\\subsection{The Quasi-Geostrophic model (QG)}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figure3.pdf}\n    \n    \\begin{minipage}[t]{.33333\\linewidth}\n    \\centering\n   \n    \\end{minipage}%\n    \\begin{minipage}[t]{.33333\\linewidth}\n    \\centering\n   \n    \\end{minipage}%\n    \\begin{minipage}[t]{.33333\\linewidth}\n    \\centering\n   \n    \\end{minipage}\n    \n    \\caption{Analysis RMSE results with the QG equations. For QG, $ M=100$, and Gaussian and Laplace samples, as compared to the LETKF with both the Gaspari-Cohn decorrelation function (GC) and our operational approximation (OP)~\\eqref{eq:operational-decorrelation}.}\n    \\label{fig:qgT1T2}\n\\end{figure*}\n\nWe follow the QG formulations given in~\\cite{san2015stabilized,mou2019data}. We discretize the equation\n\\begin{equation}\n  \\begin{split}\n    \\label{eq:QG}\n    \\omega_t + J(\\psi,\\omega) - {Ro}^{-1}\\, \\psi_x &= {Re}^{-1}\\, \\Delta\\omega + {Ro}^{-1}\\,F, \\\\\n    J(\\psi,\\omega)&\\equiv \\psi_y \\omega_x - \\psi_x \\omega_y,\n  \\end{split}\n\\end{equation}\nwhere $\\omega$ stands for the vorticity, $\\psi$ stands for the stream function, $Re$ is the Reynolds number, $Ro$ is the Rossby number, $J$ is the Jacobian term, and $F$ is a constant (in time) forcing term.\n\nThe relationship between stream and vorticity, $\\omega = -\\Delta\\psi$ is explicitly enforced in the evaluation of the ODE. The forcing term is a symmetric double gyre,\n\\begin{equation}\n  F = \\sin\\left(\\pi(y-1)\\right).\n\\end{equation}\nHomogeneous Dirichlet boundary conditions are enforced on the spatial domain $[0,1]\\times[0,2]$. The spatial discretization is a second order central finite difference for the first derivatives, and the Laplacian, with the Arakawa  approximation \\cite{arakawa1966computational} (a pseudo finite element scheme~\\cite{jespersen1974arakawa})  used for computing the Jacobian term. All spatial discretizations exclude the trivial boundary points from explicit computation.\n\nThe matrix $\\!P$ is approximated from 700 snapshots of the solution about 283 hours apart each, with Gaspari-Cohn localization applied, so as to keep the matrix sparse. The true model is run outside of time of the snapshots so as to not pollute the results. Nature utilizes a $255\\times511$ spatial discretization, and the model a $63\\times127$ spatial discretization. Observations are first relaxed into the model space via multigridding~\\cite{zubair2009efficient}, \nthen 150 distinct spatial points (using an observation operator similar to \\cite{sakov2008deterministic}) from the non-linear observation operator,\n\\begin{equation}\n    \\!H(\\psi) = \\sqrt{\\psi_x^2 + \\psi_y^2},\n\\end{equation}\nrepresenting zonal wind magnitude, are taken. The observation error is unbiased, with covariance $\\*R = 4\\mathbf{I}_{150}$. The number of synthetic ensemble members is fixed at a constant $ M = 100$, as to be more than the number of full model run ensemble members, but significantly less than the rank of the covariance.\nObservations are taken $\\Delta t=0.010886$ time units (representing one day in model space) apart.\nWe run a total of 350 assimilation steps, taking the first 50 as spinup. Results are averaged over 5 model runs (with the same nature run, but different initializations of the dynamic ensemble), with diverging runs treated as de-facto infinite error.\n\nOperationally, the LETKF assumes that observations are sufficiently far  apart that almost no two points are influenced by the same observation, and the algorithm runs on `patches' defined by the observations. In such an operational framework, equation~\\eqref{eq:rloc} cannot be applied. Thus,  using an extremely nice decorrelation function such as Gaspari-Cohn~\\cite{gaspari1999construction} (GC) is not operationally feasible. Defining sharp boundaries for the patches would be equally unfair. As such, in addition to looking at the GC decorrelation function, we will construct a decorrelation function that has both a built-in cut-off radius, but also allows for a smoother coupling between neighboring patches:\n\\begin{equation}\n    \\ell(k) = \\begin{cases}1 & k \\leq 1\\\\ {(5 - 4k)}^2{(8k - 7)} & 1 < k \\leq \\frac{5}{4}\\\\\n    0 & const.\\end{cases},\\label{eq:operational-decorrelation}\n\\end{equation}\nwhere $k_{ij} =d_{ij}\/r$ is the ratio between the distance of two given points and the decorrelation radius (which we will set to 15 grid points). The nonlinear term is an approximation to a sigmoid function.\n\n\\subsection{QG assimilation results}\nFigure~\\ref{fig:qgT1T2} reports the results with the QG model. The shrinkage approach handily beats the operational approximation decorrelation function~\\eqref{eq:operational-decorrelation} both in terms of stability and in terms of RMSE for the vast majority of chosen dynamic ensemble size $N$ and inflation factor $\\alpha$. Comparing our methodology to the LETKF with Gaspari-Cohn (GC) localization, we see that GC significantly decreases the error for larger values of $N$ and $\\alpha$, but is not stable for more operational under-sampled dynamic ensemble sizes and low inflation factors, as opposed to our shrinkage method. Possible sources of error are both the nonlinear observations and the coarse approximation to the covariance estimate. \n\nThese results lend additional support to the argument that a better (perhaps more heuristic) approximation to the shrinkage factor $\\gamma$ is needed in order to use our methodology coupled with localization to stabilize operational implementations of the LETKF.\n\nThe quasi-geostrophic results indicate that our methodology holds promise to be of use for practical data assimilation systems, especially those for which the observations are non-linear transformations of the state representation. However, the methodology needs to be refined with more optimal shrinkage factors for operational undersampled empirical covariances.\n\n\n\\begin{remark}\nAn operational implementation of the LETKF requires $m\\times N$ linear solves and $m$ matrix square roots, while our stochastic shrinkage algorithm requires $N + M$ linear solves and one matrix square root. Thus as the number of observations grows, the stochastic shrinkage methodology becomes a lot more compelling.\n\\end{remark}\n\n\n\n\n\\section{Discussion}\n\nIn~\\cite{Sandu_2015_covarianceShrinkage} it was shown that shrinkage covariance matrix estimators can greatly improve the performance of the EnKF.\nThis work develops ensemble transform filters that utilize a stochastic implementation of shrinkage covariance matrix estimates. \nWe compare our filter to the current state-of-the-art LETKF algorithm. Lorenz '96 results indicate that the new filter performs worse in the under-sampled regime than the best `static' shrinkage method, and performs better (in terms of less variance in the error) than an optimal dynamic shrinkage method for the sufficiently sampled case. Additional results with QG indicate that our method could potentially be used to augment operational LETKF implementations, but that more work is needed to devise better heuristic estimates of the shrinkage factor $\\gamma$ such that the two approaches could potentially be coupled.\n\n\n\n\\begin{acknowledgments}\nThe first two authors would like to thank Traian Iliescu and Changhong Mou for their in-depth help with understanding of the Quasi-Geostrophic model, Steven Roberts, the primary maintainer of the ODE Test Problems package, and the rest of the members of the Computational Science Laboratory at Virginia Tech for their continuous support. \n\nThe authors would like to thank Dr. Pavel Sakov and a second, Anonymous, referee for their  insightful feedback that lead to an improved paper.\n\nThe first two authors were supported, in part, by the National Science Foundation through awards NSF ACI--1709727 and NSF CCF--1613905, AFOSR through the award AFOSR DDDAS 15RT1037, and by DOE ASCR through the award  DE--SC0021313.\n\nThe last author was supported by the Research Council of Norway and the companies AkerBP, Wintershall--DEA, V{\\aa}r Energy, Petrobras, Equinor, Lundin and\nNeptune Energy, through the Petromaks--2 project (280473) DIGIRES (\\url{http:\/\/digires.no})\n\\end{acknowledgments}\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction} \n  Inflationary scinario solves the problems of the flatness, horizon, and the origin of the fluctuation at the same time. \nOne of the remarkable prediction of inflation is the power spectrum of the cosmic microwave back-ground (CMB). \nThese prediction has been confirmed in great accuracy and provide severe constraints on the inflationary models\\cite{Adeetal2016a}.\n  \n  However, it has been pointed out that there is a discrepancy of the Hubble parameters \nif we assume the flat-$\\rm{\\Lambda}$CDM model\\cite{Sahnietal2014,Dingetal2015,Zhengetal2016}. \nAnother problem which may be related with the Hubble constant is that\nthe CMB data favor the value $H_0 = (67.6 \\pm 0.6) \\rm{km s}^{-1}\\rm{Mpc}^{-1}$\\cite{Adeetal2016a} \nwhereas the local measurement favors $H_0 = (73.24 \\pm 1.74) \\rm{km s}^{-1}\\rm{Mpc}^{-1}$\\cite{Riessetal}. \nSince we do not know the reason of this discrepancy, it may be wise to find other models.\n\n  In the inflationary models, the seed of the fluctuation is the quantum theory of the inflaton. \nThe quantum fluctuation of inflaton or some other scalar fields related to inflation (for example, the fields in the hybrid inflation) freezes to some classical value through the exponential expansion of the universe. \nIn the theoretical side, it is known that there is an ambiguity in the choice of the vacuum. \nFor example, in de-Sitter spacetime, it is known that there is no de-Sitter invariant vacuum so we have to choose a vacuum state. \nOne natural choice is the vacuum state associated with the mode function which reduces to the ordinary positive frequency mode in Minkowski spacetime\nin the past infinity.\nNamely, the mode function $v_k$ is chosen by the requirement for the \"in-state\":\n  \\begin{equation}\n  \\lim_{\\eta \\rightarrow -\\infty}v_{\\omega}(\\eta)=\\frac{1}{\\sqrt{2\\omega}}e^{-i\\omega \\eta}\n  \\end{equation}\n  This requirement fixes our vacuum state in de-Sitter expansion era. \nHowever if we have pre-inflation era, the past infinity of the inflation era may be affected by the history of the pre-inflation evolution.\nAssuming that the inflation is well below the Planck scale, we expect the pre-inflation evolution of the universe is dominated by radiation as well as the effects of spatial curvature. \nNote that if our universe started with the quantum to classical transition, \nwe naturally expect that the kinetic energy of the space is the same order as the potential energy caused by the spatial curvature, which is the origin of the flatness problem. \n\n\nThe effect of the spatial curvature for density perturbation in inflation has been analyzed in several papers \\cite{RatraPeebles1994,RatraPeebles1995,Bucheretal1995,LythWoszczyna1995,Yamamotoetal1995}. \nThey showed that the power spectrum for low $l$ region deviates from flat spectrum.\nThe analysis comparing to Plank results has been done in Ref.\\cite{BGY, ORS}.\nHowever, there is the  possibility that the radiation in the pre-inflation era also affects the choice of the vacuum. \nUsually we expect that they affects the overall normalization as a transfer\nfunction.\n\nIn this paper, we consider the effect of both the radiation and the curvature, and will show the exact solution of the inflaton (massless scalar field) equation. \nBy quantizing the inflaton, we will derive the exact power spectrum and show that it is almost flat expect for the very large scale. \nIn the visible scale, the modification seems to be very small but may be observable.\n\nIn the next section, we will consider the pre-inflation era and will find that the scale factor is written by Weierstrass elliptic function in conformal time.\nIn section 3, we will show that the field equation of the inflaton can be written as Lam\\'e equation so that the solutions can be written by elliptic functions. \nBy using various formulas concerning elliptic functions, we derive the exact power spectrum.\nThe final section will be devoted to the discussions.\n\n \n\\section{Pre-inflation era}\n\nWe consider the Friedmann-Robertson-Walker metric in conformal time\n\\begin{equation}\n\tds^2=a^2(\\eta)\\left[-d\\eta^2+\\frac{dr^2}{1-Kr^2}+r^2d^2\\Omega \\right].\n\\end{equation}\nWe do not know much about pre-inflation era. The inflation may start from quantum to classical transition in gravity. \nHowever, inflaton may be related to the Grand Unified Theories where the energy scale may be lower than the Planck scale. \nTherefore, it is reasonable to think that our universe began with many relativistic matters. \nIf our universe started with the quantum to classical transition, it is reasonable to expect that the kinetic energy of our universe, \nwhich is related to expansion rate, and the potential energy, which is related to the spatial curvature, are the same order.  \nThe spatial curvature is suppressed during inflation to resolve the flatness problem.\nIn the inflationary scenario, the vacuum energy caused by the potential is the origin of the inflation era. \nThere are many interpretation of the origin of the potential. \nIn the chaotic inflation, stochastic process is the origin of the initial value of the potential. \nBut here we assume that even before the inflation, the energy caused by the effective potential is present whose value is denoted by $V_0(>0)$.\nThen the Friedmann equation is given by\n\\begin{equation}\n\t\\left(\\frac{1}{a^2}\\frac{da}{d\\eta}\\right)^2=\\frac{8\\pi G}{3}\\rho-\\frac{K}{a^2},\n\\end{equation}\nwhere $K$ is the curvature of the space and $a'=da\/d\\eta$. The energy density is dominated by the radiation. \nTherefore, we include the radiation density as well as the inflaton potential:\n\\begin{equation}\n\t\\rho=\\frac{\\rho_r}{a^4}+V_0.\n\\end{equation}\nWe assume that quantum to classical transition occured at Plank scale and the curvature energy is almost the same order of energy of the radiation. \nTherefore, after the quantum to classical transition of the space-time, our universe is radiation and curvature dominant followed by the era of\nvacuum energy dominant. Exponential growth starts when\n\\begin{equation}\n\t\\frac{|K|}{a^2}\\sim\\frac{8\\pi G}{3}V_0.\n\\end{equation}\n\nWe will use the following notation\n\\begin{equation}\n\t\\frac{8\\pi G V_0}{3}= H^2, \\quad A=-\\frac{K}{H^2},\\quad B=\\frac{\\rho_r}{V_0}.\n\\end{equation}\nThen we have \n\\begin{equation}\n\tH\\eta=\\int^a\\frac{da}{(a^4+Aa^2+B)^{1\/2}}\\label{eq:defeta}.\n\\end{equation}\n$a^4+Aa^2+B=0$ has two solutions, $a^2=-A\/2+\\sqrt{(A\/2)^2-B}$ and $a^2=-A\/2-\\sqrt{(A\/2)^2-B}$, which will be denoted by $\\tilde{e}_2,\\tilde{e}_3$, respectively.\nThere are two cases: (i) $A>-2\\sqrt{B}$, and (ii) $A<-2\\sqrt{B}$.\nFor the case (i), the universe can start from $a=0$ since the singularities in the integrand of (\\ref{eq:defeta}) are not on the real axis.  \nTherefore we can fix the integration constant as \n\\begin{equation}\n\tH\\eta=\\int_0^a\\frac{da}{(a^4+Aa^2+B)^{1\/2}}=\\frac{1}{2}\\int_0^{a^2}\\frac{dx}{(x^3+Ax^2+Bx)^{1\/2}},\n\\end{equation}\nwhere $x=a^2$.\nBy shifting integration variable as $x=y-A\/3$, we can remove the quadratic term,\n\\begin{equation}\n\tH \\eta=\\int_{e_1}^{a^2+e_1}\\frac{dy}{[4(y-e_1)(y-e_2)(y-e_3)]^{1\/2}},\\label{yint}\n\\end{equation}\nwhere \n\\begin{equation}\n\te_1=\\frac{A}{3},\\qquad e_2=\\tilde{e}_2+\\frac{A}{3},\\qquad e_3=\\tilde{e}_3+\\frac{A}{3},\n\\end{equation}\nwhich satisfy\n\\begin{equation}\n\te_1+e_2+e_3=0.\n\\end{equation}\nSince (\\ref{yint}) can be decomposed as\n\\begin{equation}\n\tH\\eta= \\int_{e_1}^\\infty\\frac{dy}{[4(y-e_1)(y-e_2)(y-e_3)]^{1\/2}}-\\int_{a^2+e_1}^\\infty\\frac{dy}{[4(y-e_1)(y-e_2)(y-e_3)]^{1\/2}},\n\\end{equation}\nwe can write the inverse relation by using the Weierstrass elliptic function as follows:\n\\begin{equation}\n\ta(\\eta)=[\\wp(\\omega_1-{\\tilde{\\eta}})-e_1]^{1\/2},\\label{eq:aone}\n\\end{equation}\nwhere $\\wp$ is defined as\n\\begin{equation}\n\t\\wp(z)=\\frac{1}{z^2}+\\sum_{(m,n)\\neq(0,0)}\\left[\\frac{1}{(z-2m\\omega_1-2n\\omega_2)^2}-\\frac{1}{(2m\\omega_1+2n\\omega_2)^2}\\right],\n\\end{equation}\nand\n\\begin{equation}\n\t{\\tilde{\\eta}}=H\\eta.\n\\end{equation}\n$\\omega_1$ is one of the half periods and given by\n\\begin{equation}\n\t\\omega_1=\\int_{e_1}^\\infty\\frac{dy}{[4(y-e_1)(y-e_2)(y-e_3)]^{1\/2}}.\n\\end{equation}\nIt is easy to see that for small $\\eta$ we have\n\\begin{equation}\n\ta(\\eta)\\sim \\sqrt{\\tilde{e}_2\\tilde{e}_3}{\\tilde{\\eta}}=\\sqrt{B}{\\tilde{\\eta}},\n\\end{equation}\nwhereas $a$ approaches\n\\begin{equation}\n\ta(\\eta)\\sim \\frac{1}{\\omega_1-{\\tilde{\\eta}}},\n\\end{equation}\nwhen $\\eta \\rightarrow \\omega_1\/H$, which represents de-Sitter phase in conformal time.\n\nFor the case (ii), \nthe integrand of (\\ref{eq:defeta}) has two singularities on the positive real axis at $a=\\tilde{e}_2$ and $a=\\tilde{e}_3(<\\tilde{e}_2)$.\nThus, the universe starts with a finite value $a=\\sqrt{\\tilde{e}_2}$ and our universe does not have the ``initial singularity''.\nIn this case we have\n\\begin{equation}\n\ta(\\eta)=[\\wp(\\omega_2-{\\tilde{\\eta}})-e_1]^{1\/2}, \\label{eq:atwo}\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\omega_2=\\int_{e_2}^\\infty\\frac{dy}{[4(y-e_1)(y-e_2)(y-e_3)]^{1\/2}}\n\\end{equation}\nis another half period.\nBehavior around $\\eta\\sim0$ is different from that of case (i),\n\\begin{equation}\n\ta(\\eta)\\sim\\sqrt{\\tilde{e}_2}\\left(1+\\frac{\\tilde{e}_2-\\tilde{e}_3}{2}{\\tilde{\\eta}}^2\\right),\n\\end{equation}\nwhile de-Sitter phase appears around ${\\tilde{\\eta}}\\sim\\omega_2$,\n\\begin{equation}\n\ta(\\eta)\\sim\\frac{1}{\\omega_2-{\\tilde{\\eta}}}.\n\\end{equation}\n\n\n\\section{Exact solution of massless scalar fields and the spectrum of the density perturbation}\nIn this section, we solve the equation of massless scalar field exactly on the background spacetime derived in the previous section, \nand quantize it to calculate the power spectrum.\nThe equation of massless scalar field $\\psi$ is given by\n\\begin{equation}\n\t\\frac{\\partial^2}{\\partial \\eta^2}\\psi(x,\\eta)+\\frac{2}{a}\\frac{da}{d\\eta}\\frac{\\partial}{\\partial \\eta}\\psi(x,\\eta)-{\\Delta}\\psi(x,\\eta)=0.\n\t\\label{fieldequation}\n\\end{equation}\nIf we decompose the solution as $\\psi(x,\\eta)=\\chi_k(\\eta)\\phi_k(x)$, where $\\Delta\\phi_k(x)=-k^2\\phi_k(x)$,\nthe equation for $\\chi_k(\\eta)$ is\n\\begin{equation}\n\\frac{d^2}{d \\eta^2}\\chi_k+\\frac{2}{a}\\frac{da}{d\\eta}\\frac{d}{d \\eta}\\chi_k+k^2\\chi_k=0.\n\\end{equation}\nWe shall use the variable ${\\tilde{\\eta}}=H\\eta$ and write the above equation as\n\\begin{equation}\n\\chi^{\\prime\\prime}_k+2\\frac{a^\\prime}{a}\\chi^\\prime_k+\\tilde{k}^2\\chi_k=0,\n\\end{equation}\nwhere the prime denotes the derivative with respect to ${\\tilde{\\eta}}$ and $\\tilde{k}=k\/H$.\nBy rescaling $\\chi_k$ as\n\\begin{equation}\nv_k=a\\chi_k,\n\\end{equation}\nwe have the following equation for $v_k$:\n\\begin{equation}\nv^{\\prime\\prime}_k+\\left(-\\frac{a^{\\prime\\prime}}{a}+\\tilde{k}^2\\right)v_k=0.\n\\end{equation}\nInserting (\\ref{eq:aone}) for the case (i), we find\n\\begin{equation}\n\\frac{d^2}{d{\\tilde{\\eta}}^2}v_k=\\left[2\\wp(\\omega_1-{\\tilde{\\eta}})+e_1-\\tilde{k}^2\\right]v_k\\label{eq:eqv}\n\\end{equation}\nThe equation for the case (ii) is quite similar. We obtain\n\\begin{equation}\n\\frac{d^2}{d{\\tilde{\\eta}}^2}v_k=\\left[2\\wp(\\omega_2-{\\tilde{\\eta}})+e_1-\\tilde{k}^2\\right]v_k.\n\\end{equation}\n\nWe observe that these equations are the Lam\\'{e} equation \n\\begin{equation}\n\\frac{d^2}{dx^2}y(x)=\\left[l(l+1)\\wp(x)+h\\right]y(x),\n\\end{equation}\nwith $l=1, h=e_1-\\tilde{k}^2$.\nThe solution of the Lam\\'{e} equation for $l=1$ is a classical result\\cite{RLH}\\footnote{The method used in Ref. \\cite{RLH} has been applied\nto the evolution equation for gravitational waves in Ref. \\cite{Exact}}. For case (i), two independent solutions are\n\\begin{align}\nv_k&=a_0\\frac{\\sigma(\\omega_1-{\\tilde{\\eta}}+c)}{\\sigma(\\omega_1-{\\tilde{\\eta}})\\sigma(+c)}e^{-(\\omega_1-{\\tilde{\\eta}})\\zeta(+c)},\\notag\\\\\nv_{-k}&=a_0\\frac{\\sigma(\\omega_1-{\\tilde{\\eta}}-c)}{\\sigma(\\omega_1-{\\tilde{\\eta}})\\sigma(-c)}e^{-(\\omega_1-{\\tilde{\\eta}})\\zeta(-c)},\\label{eq:exactsolutions1}\n\\end{align}\nwhere $a_0$ is the normalization constant, and $\\zeta(z)$ and $\\sigma(z)$ are defined as \\cite{HTF}\n\\begin{align}\n&\\zeta(z)=\\frac{1}{z}+\\sum{}^\\prime\\left[\\frac{1}{z-\\omega}+\\frac{1}{\\omega}+\\frac{z}{\\omega^2}\\right], \\\\\n&\\sigma(z)=z\\prod{}^\\prime\\left[\\left(1-\\frac{z}{\\omega}\\right)\\exp\\left(\\frac{z}{\\omega}+\\frac{1}{2}\\frac{z^2}{\\omega^2}\\right)\\right],\n\\end{align}\nwhere $\\sum'=\\sum_{(m,n)\\neq(0,0)}, \\prod'=\\prod_{(m,n)\\neq(0,0)}$, and $\\omega=2m\\omega_1+2n\\omega_2$.\nThey are related to the Weierstrass elliptic function as follows:\n\\begin{equation}\n\t\\wp(z)=-\\zeta'(z),\\ \\ \\zeta(z)=\\frac{\\sigma'(z)}{\\sigma(z)}.\n\\end{equation}\nThe value of $c$ is defined as\n\\begin{equation}\n\\wp(c)=e_1-\\tilde{k}^2.\\label{eq:cdef1}\n\\end{equation}\nExpansion around $z=0$ can be derived as\n\\begin{align}\n\\wp(z)&=\\frac{1}{z^2}+\\cdots,\\notag\\\\\n\\zeta(z)&=\\frac{1}{z}+\\cdots,\\notag\\\\\n\\sigma(z)&=z+\\cdots.\\label{eq:aroudorigin}\n\\end{align}\nWe also point out that $\\wp(z)$ is an even function whereas $\\zeta(z),\\sigma(z)$ are odd functions.\nAlthough the equation (\\ref{eq:cdef1}) determines $c$ only up to the periods of $\\wp(z)$\n\\footnote{There is another ambiguity because $\\wp(-c)=\\wp(c)$. Replacing $c$ with $-c$ corresponds to the change $v_k\\leftrightarrow v_{-k}$. \nWe will fix this ambiguity later (see (\\ref{wpprime})).},\nthe solutions (\\ref{eq:exactsolutions1}) are not ambiguous because these are periodic functions with respect to $c$,\nnamely \n\\begin{equation}\nv_k(c+2\\omega_i)=v_k(c)\\label{eq:periodicc}.\n\\end{equation}\nThis result can be derived by using quasi-periodic properties\n\\begin{equation}\n\\zeta(z+2\\omega_i)=\\zeta(z)+2\\eta_i,\\qquad \\sigma(z+2\\omega_i)=-\\sigma(z)\\exp[2(z+\\omega_i)\\eta_i],\n\\end{equation}\nwhere $\\eta_i=\\zeta(\\omega_i)$.\n\nFor the case (ii), the solutions are obtained as\n\\begin{align}\nv_k&=a_0\\frac{\\sigma(\\omega_2-{\\tilde{\\eta}}+c)}{\\sigma(\\omega_2-{\\tilde{\\eta}})\\sigma(c)}e^{-(\\omega_2-{\\tilde{\\eta}})\\zeta(c)},\\notag\\\\\nv_{-k}&=a_0\\frac{\\sigma(\\omega_2-{\\tilde{\\eta}}-c)}{\\sigma(\\omega_2-{\\tilde{\\eta}})\\sigma(-c)}e^{(\\omega_2-{\\tilde{\\eta}})\\zeta(c)},\\label{eq:exactsolutions2}\n\\end{align}\nwhich are also periodic with respect to $c$.\n\nLet us next prove that $v_k$ and $v_{-k}$ are complex conjugates when $B>A^2\/4$ (included in case (i)).\nBy (\\ref{eq:cdef1}), we find \n\\begin{align}\nc&=\\frac{1}{2}\\int_{0}^\\infty\\frac{dx}{x^{1\/2}(x^2+Ax+B)^{1\/2}}+\\frac{1}{2}\\int_{-\\tilde{k}^2}^0\\frac{dx}{x^{1\/2}(x^2+Ax+B)^{1\/2}}\\notag\\\\\n&=\\omega_1+\\frac{1}{2}\\int_{-\\tilde{k}^2}^0\\frac{dx}{x^{1\/2}(x^2+Ax+B)^{1\/2}}. \\label{cint}\n\\end{align}\nThe second term is pure imaginary so that\n\\begin{equation}\n(c-\\omega_1)^*=-(c-\\omega_1),\n\\end{equation}\nwhich leads to\n\\begin{equation}\n\tc^*=2\\omega_1-c\\sim-c, \\label{cstar}\n\\end{equation}\nwhere ``$\\sim$'' denotes the equivalence up to the periods.\nNote that we also used the fact that $\\omega_1$ is real for $B>A^2\/4$.\nThen,\nit is straightforward to prove\n\\begin{equation}\nv_k^*=v_{-k}, \\ \\ {}^\\forall k>0\n\\end{equation}\nThis relation also holds for $A<-2\\sqrt{B}$ (case (ii)).\n\nIn the case $A>2\\sqrt{B}$, on the other hand, the relation between $v_k$ and $v_{-k}$ depends on $k$.\nThis is because complex conjugate of $c$ behaves differently from (\\ref{cstar}) as follows:\n\\begin{equation}\n\tc^*\\sim\n\t\\begin{cases}\n\t\tc \\ \\ (\\tilde{k}^4-A\\tilde{k}^2+B<0) \\\\\n\t\t-c \\ \\ (\\tilde{k}^4-A\\tilde{k}^2+B>0).\n\t\\end{cases}\n\\end{equation}\nThis difference can be seen from the second term in the equation (\\ref{cint}), whose integrand becomes real near $x=-\\tilde{k}^2$ leading to $c^*\\sim c$.\nIt follows then $v_k$ is real for the wave number $k$ such that $c^*(k)\\sim c(k)$,\n\\begin{equation}\n\tv_k^*=v_k, \\ \\ \\tilde{k}^4-A\\tilde{k}^2+B<0.\n\\end{equation}\nThis result shows that the wave function $v_k$ is deformed in this region so that $v_k$ can no longer be regarded as a mode function to quantize.\nFor this reason, we concentrate on the case $A<2\\sqrt{B}$ in the rest of this paper.\n\n\nWe are going to find the the normalization of the solutions. \nWe will use the following normalization:\n\\begin{equation}\nv_k\\frac{d}{d\\eta}v_k^*-v_k^*\\frac{d}{d\\eta}v_k=i,\\label{eq:normalizationcondition}\n\\end{equation}\nThis normalization is equivalent to considering $v_k\\sim e^{-ik\\eta}\/\\sqrt{2k}$ for massless scalar field in flat space.\nBy explicit evaluation of (\\ref{eq:exactsolutions1}) and (\\ref{eq:exactsolutions2}) using the following formulas\\cite{HTF}, \n\\begin{align}\n\\sigma(u-v)\\sigma(u+v)&=-\\sigma^2(u)\\sigma^2(v)[\\wp(u)-\\wp(v)],\\notag\\\\\n\\zeta(u+v)&=\\zeta(u)+\\zeta(v)+\\frac{1}{2}\\frac{\\wp^\\prime(u)-\\wp^\\prime(v)}{\\wp(u)-\\wp(v)},\\label{eq:additionaltheorems}\n\\end{align}\nwe find \n\\begin{equation}\n\tv_k\\frac{dv^*_k}{d\\eta}-v^*_k\\frac{dv_k}{d\\eta}=-a_0^2H\\wp'(c).\n\\end{equation}\n$\\wp'(c)$ is determined up to sign by the differential equation $(\\wp'(z))^2=4(\\wp(z)-e_1)(\\wp(z)-e_2)(\\wp(z)-e_3)$ with the definition of $c$ (\\ref{eq:cdef1}).\nHere we take \n\\begin{equation}\n\t\\wp'(c)=-2i\\tilde{k}\\sqrt{\\tilde{k}^4-A\\tilde{k}^2+B} \\label{wpprime}\n\\end{equation}\nto ensure that $v_k$ represents the positive frequency mode around $\\eta\\sim0$ as will be shown in the \nnext paragraph.\nThen, the normalization condition (\\ref{eq:normalizationcondition}) gives\n\\begin{equation}\n\ta_0=\\frac{1}{\\sqrt{2\\tilde{k} H}(\\tilde{k}^4-A\\tilde{k}^2+B)^{1\/4}}.\n\\end{equation}\n\nBefore considering the power spectrum, we must choose the vacuum state of the quantum field.\nTo do so,\nwe first derive the behavior of the mode function $v_k(\\eta)$ in the past infinity. The past infinity corresponds to ${\\tilde{\\eta}}=0$.\nWe rewrite $v_k(\\eta)$ as\n\\begin{align}\nv_k(\\eta)\/v_k(0)&=\\exp\\left[\\ln\\sigma(\\omega_1-{\\tilde{\\eta}}+c)-\\ln\\sigma(\\omega_1+c)-(\\ln\\sigma(\\omega_1-{\\tilde{\\eta}})-\\ln\\sigma(\\omega_1))+{\\tilde{\\eta}}\\zeta(c)\\right],\\notag\\\\\n&=\\exp\\left[\\int_{\\omega_1}^{\\omega_1-{\\tilde{\\eta}}}(\\zeta(x+c)-\\zeta(x)-\\zeta(c))dx\\right]. \\label{vk}\n\\end{align}\nBy using (\\ref{eq:additionaltheorems}), we have\n\\begin{align}\nv_k(\\eta)\/v_k(0)&=\\exp\\left[\\frac{1}{2}\\int_{\\omega_1}^{\\omega_1-{\\tilde{\\eta}}}\\frac{\\wp^\\prime(x)-\\wp^\\prime(c)}{\\wp(x)-\\wp(c)}dx\\right]\\notag\\\\\n&=\\left(\\frac{\\wp(\\omega_1-{\\tilde{\\eta}})-\\wp(c)}{\\wp(\\omega_1)-\\wp(c)}\\right)^{1\/2}\n\t\\exp\\left[\\frac{-\\wp^\\prime(c)}{2}\\int_{\\omega_1}^{\\omega_1-{\\tilde{\\eta}}}\\frac{dx}{\\wp(x)-\\wp(c)}\\right]. \\label{vk2}\n\\end{align}\nWe evaluate (\\ref{vk2}) for ${\\tilde{\\eta}}<\\omega_1$. Since $\\wp(\\omega_1-{\\tilde{\\eta}})=e_1+\\rm{O}({\\tilde{\\eta}}^2)$,\nwe obtain \n\\begin{equation}\n\tv_k(\\eta)\/v_k(0)\\sim \\exp\\left[-i\\tilde{k}(1-A\/\\tilde{k}^2+B\/\\tilde{k}^4)^{1\/2}{\\tilde{\\eta}}\\right] \\label{pastbehavior}\n\\end{equation} for small ${\\tilde{\\eta}}$.\nThis result shows that, in the past, $v_k$ behaves as the mode function in the flat spacetime, i.e. $e^{-ik\\eta}$,  only for large $k$ while the wave number is deformed\nfor small $k$. So we fix the mode function by considering large $k$ behavior. \n\nWe expand the quantum field as\n\\begin{equation}\n\t\\chi(\\eta,x)=\\frac{1}{a(\\eta)}\\sum_k\\left(a_kv_k(\\eta)\\phi_k(x)+a^\\dagger_kv_k^*(\\eta)\\phi^*_k(x)\\right).\n\\end{equation}\nThis estimate of the asymptotic behavior is consistent with the normalization by (\\ref{eq:normalizationcondition}).\nWe are considering large $\\tilde{k}$ region, where flat space approximation is valid, therefore\n\\begin{equation}\n\\chi(\\eta,x)=\\frac{1}{a}\\int\\frac{d^3k}{(2\\pi)^{3\/2}}(a_{\\bf{k}} v_k(\\eta)e^{i\\bf{k}\\cdot\\bf{x}}+a_{\\bf{k}}^\\dagger v_k^*(\\eta)e^{-i\\bf{k}\\cdot\n\t\\bf{x}})=\\int\\frac{d^3k}{(2\\pi)^{3\/2}}(a_{\\bf{k}}\\chi_k+a^\\dagger_{\\bf{k}}\\chi^*_k).\n\\end{equation}\nBy using the explicit solutions (\\ref{eq:exactsolutions1}) and (\\ref{eq:exactsolutions2}), we get\n\\begin{equation}\n\t\\chi^*_k\\chi_k=\\frac{v_k^*v_k}{a^2}=\\frac{H^2}{2k^3(1-AH^2\/k^2+BH^4\/k^4)^{1\/2}}\\left(1+\\frac{k^2}{H^2a^2}\\right).\n\\end{equation}\nAfter inflation $(a\\gg1)$, this value is frozen to\n\\begin{equation}\n\\chi_k^*\\chi_k\\to\\frac{H^2}{2k^3(1-AH^2\/k^2+BH^4\/k^4)^{1\/2}}.\n\\end{equation}\nBy the usual definition of the power spectrum\n\\begin{equation}\n\\mathcal{P}_\\chi(k)=\\frac{k^3}{2\\pi^2}\\chi_k^*\\chi_k,\n\\end{equation}\nwe finally obtain the following power spectrum:\n\\begin{equation}\n\\mathcal{P}_\\chi(k)=\\left(\\frac{H}{2\\pi}\\right)^2\\frac{1}{(1-AH^2\/k^2+BH^4\/k^4)^{1\/2}}.\n\\end{equation}\n\nOne of the prediction of this spectrum is that al large scale {\\it i.e.} sufficiently small $k$, the perturbation spectrum goes to zero whereas it goes to constant value at large $k$.\nSmall $k$ behavior is understood from (\\ref{pastbehavior}). If we introduce the effective wave number $q(k)=k(1-A\/\\tilde{k}^2+B\/\\tilde{k}^4)^{1\/2}$ to write\n$v_k(\\eta)\/v_k(0)\\sim e^{-iq\\eta}$, we can see that $q(k)$ has the minimum $q_{\\rm{min}}=H\\sqrt{2\\sqrt{B}-A}$ at $\\tilde{k}=B^{1\/4}$.\nThus, the radiation energy is a kind of infrared cutoff. As a result, the vacuum expectation value of $\\chi^2$, which is evaluated as the integral\nof $\\mathcal{P}_\\chi(k)\/k$, is IR convergent in contrast to the usual de-Sitter vacuum case.\n\nWhen the curvature is negartive ($K<0 \\Leftrightarrow A>0$), there appears an enhancement of the perturbation at small $k$.\nAs an example, we list a figure (Fig.\\ref{fig:solutions1}) for open space $(K<0)$ for the values $A=5\\times 10^{-3},B=2\\times 5^4\\times 10^{-8}$.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[clip,width=10.0cm]{densityperturbation.pdf}\n\\caption{A plot of the spectrum of $\\mathcal{P}_\\chi(k)$ normalized by $(H\/2\\pi)^2$ for open universe $A=5\\times 10^{-3},B=2\\times 5^4\\times 10^{-8}$ at very high super horizon wavelength. We can see an enhancement of the power spectrum for small $k$.}\n\\label{fig:solutions1}\n\\end{center}\n\\end{figure}\nFor this parameter, we find that there is a very small deviation from flat space and there is a peak at $k=H\\sqrt{2B\/A}$, which may be invisible since the length scale is too large. \nHowever, if we consider closed universe ($K>0$), there is no enhancement but monotonically decrease as $k$ becomes smaller (Fig.\\ref{fig:solutions2}). \n\\begin{figure}[h\n\\begin{center}\n\\includegraphics[clip,width=10.0cm]{densityperturbation2.pdf}\n\\caption{A plot of the spectrum $\\mathcal{P}_\\chi(k)$ for closed universe $A=-5\\times 10^{-3},B=2\\times 5^4\\times 10^{-8}$ at very high super horizon wavelength. We see no enhancement of power spectrum for small $k$.}\n\\label{fig:solutions2}\n\\end{center}\n\\end{figure}\n\n\\section{Summary and Discussions}\nThe usual inflationary scenarios assume that inflaton starts from the de-Sitter vacuum in the past infinity. \nWe here considered that we have radiation and curvature dominant era before inflation. \nThese stages affects the in-state vacuum compared with the case of usual inflation.\nWe have shown that the free scalar field equation (\\ref{fieldequation}) in this scenario can be written as Lam\\'e equation (\\ref{eq:eqv}) and can be solved exactly. \nThe solution can be written in terms of Weierstrass elliptic functions and we showed the exact power spectrum of the inflation. \nIt modifies the usual scaling behavior, especially for small $k$. \nAlthough the effect of the modification seems very small, it is interesting that the scalar field equation can be written as Lam\\'{e} equation and we could find the solution exactly. \n\nThere are some problems, however. \nOne is our assumption that the inflaton potential is present as constant even before inflation.\nThere are many scenarios for inflation, in some of which the vacuum energy happens as phase transition. \nFor such a case, we have to consider effective potential before inflation which may change in accordance with the energy scale. \nAnother problem is that we do not know whether it is valid to use free inflaton before inflation. \nThe interaction may change the behavior of the spectrum. However, it is still interesting that the free scalar field can be solved exactly.\nIt is also interesting that at sufficient value of $l$, the power spectum of CMB is almost constant but it has also enhancement for small $l$ and it looks like going to zero when $l$ is very small, although the error is still large enough.\n\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nM.Verbitsky in the work \\cite{verb} showed that for a hyperholomorphic vector bundle $F$ on a hyperk\\\"ahler manifold $X$ there are no obstructions for stable deformations of $F$ besides the Yoneda pairing on $\\Ext^1(F, F)$. Moreover, he proved the existence of a canonical hyperk\\\"ahler structure on the reduction of the coarse moduli space of stable deformations of $F$. If $S$ is a K3 surface then it is known that the Hilbert scheme $S^{[n]}$ is a hyperk\\\"ahler manifold and the tangent bundle $T_{S^{[n]}}$ is a hyperholomorphic bundle on $S^{[n]}$. Thus, the investigation of the deformation space of the bundle $T_{S^{[n]}}$ is a very natural and interesting question from the point of view of hyperkahler geometry. This question also appeared in \\cite{charles} in the context of the Lefschetz standard conjecture for hyperk\\\"ahler manifolds. It was mentioned there without a proof that for $n = 2$ the tangent bundle might actually be rigid. In the present note we confirm this statement by proving the following theorem.\n\\begin{theorem}\n\\label{main} \nLet $X$ be a manifold of K$3^{[2]}$-type. Then the tangent bundle $T_X$ is infinitesimally rigid, i.e. $H^1(X, {\\mathcal End}(T_X))=0$. \\end{theorem}\nThe proof of this statement is given in sections 3 and 4. It follows from explicit computations in the case when $X$ is the second Hilbert scheme of a K3 surface and then from application of general results from the theory of hyperholomorphic bundles \\cite{verb}. For $n>2$ the question about deformations of $T_{S^{[n]}}$ seems to be much more difficult due to more complicated geometry of the corresponding Hilbert scheme and thus should be considered separately.\n\n\\medskip\n\\noindent\\textbf{Acknowledgements.} The author is grateful to Christopher Brav and Misha Verbitsky for helpful discussions and to Fran\\c{c}ois Charles for suggestions.\n\\section{Hilbert square}\nFor a smooth projective surface $S$ the Hilbert scheme of length-2 subschemes of $S$ is denoted by $S^{[2]}$. Let $\\Delta:S\\hookrightarrow S\\times S$ be the diagonal embedding, $p_1,p_1':S\\times S\\to S$ be the projections onto the first and the second component and $\\sigma:Z \\stackrel{\\mathsf{def}}= \\mathsf{Bl}_\\Delta(S\\times S)\\to S\\times S$ be the blowup of $S\\times S$ in $\\Delta$. The natural action of the symmetric group $\\mathfrak{S}_2$ on $S\\times S$ extends to an action on $Z$ and the Hilbert square $S^{[2]}$ is the quotient of $Z$ by this action. By $q_2$ we denote the corresponding quotient map $Z\\to S^{[2]}$. Let $j:E\\hookrightarrow Z$ be the exceptional divisor of $\\sigma$. Recall that $E\\cong\\mathbb{P}(T_S)$ is a projective bundle over $S$ and we have the relative Euler exact sequence\n\\begin{equation}\\label{eul}\n0\\to\\Omega_{E\/S}\\to\\pi^*\\Omega_S(-1)\\to\\mathcal{O}_E\\to0.\n\\end{equation}\nAlso, by $\\iota:D\\hookrightarrow S^{[2]}$ we denote the isomorphic image of $E$ by $q_2$. The divisor $D$ is precisely the locus parametrizing non-reduced subschemes of $S$ of length two.\n\nPut $q_1 := p_1\\circ\\sigma$ and $q_1' := p_1'\\circ\\sigma$. The following diagram depicts the relationship among all the natural maps between the varieties that we mentioned:\n\\begin{equation}\n\\label{equation:notationHilb2}\n\\begin{tikzcd}\n& E\\cong\\mathbb{P}(T_S) \\arrow[r, hook, \"j\"] \\arrow[dl, swap, \"\\pi\"] & Z \\arrow[dl, swap, \"\\sigma\"] \\arrow[dr, \"q_2\"] \\arrow[d, \"q_1\"]\\\\ \nS \\arrow[r, hook, \"\\Delta\"] & S\\times S \\arrow[d, \"p_1'\"] \\arrow[r, \"p_1\"] & S & S^{[2]} & D. \\arrow[l, hook', swap, \"\\iota\"]\\\\\n& S &\n\\end{tikzcd}\n\\end{equation}\nHere $\\pi:E\\to S$ is the projective bunde map and the equality $\\sigma\\circ j = \\Delta\\circ\\pi$ yields $q_1\\circ j = \\pi$.\n\nNote that $Z$ is isomorphic to the universal closed subscheme $\\mathcal{Z}: = \\{(x, \\xi)\\, |\\, x\\in\\mathsf{Supp}(\\xi)\\}$ in $S\\times S^{[2]}$ and for any coherent sheaf $F$ over $S$ there is the incidence exact sequence\n\\begin{equation}\\label{incidence}\n0\\to q_1^*F(-E)\\to q_2^*F^{[2]}\\to q_1'^{*}F\\to0,\n\\end{equation}\nwhere $F^{[2]}$ is the image of $F$ under the tautological functor $q_{2*}q_1^*:\\mathsf{Coh}(S)\\to\\mathsf{Coh}(S^{[2]})$ (see \\cite[p. 193]{lehn}).\n\nRecall that $q_{2*}\\mathcal{O}_{Z}\\cong\\mathcal{O}_{S^{[2]}}\\oplus L^{-1}$, where the line bundle $L^{-1}$ is the eigenspace to the eigenvalue $-1$ of\nthe cover involution. Moreover, $L^{\\otimes2}\\cong\\mathcal{O}_{S^{[2]}}(D)$ and $q_2^* L\\cong\\mathcal{O}_Z(E)$. Note that $q_{2*}j_*\\mathcal{O}_E\\cong\\iota_*\\mathcal{O}_D$ and $q_2^*\\iota_*\\mathcal{O}_D\\cong\\mathcal{O}_{2E}$.\n\nThere is an exact sequence\n\\begin{equation}\n\\label{pullbackCotangent}\n0\\to q_2^*\\Omega_{S^{[2]}}(E)\\to\\Omega_Z(E)\\to j_*\\mathcal{O}_E\\to0\n\\end{equation}\nand an isomorphism\n\\[\\Omega_{S^{[2]}}\\cong q_{2*}(\\mathcal{N}^\\vee_{Z\/S\\times S^{[2]}}(E)).\\]\nThe sequence \\eqref{pullbackCotangent} implies that $\\omega_{q_2}\\cong\\mathcal{O}_Z(E)$, hence the right adjoint functor to $q_{2*}$ is $q_2^!(-)=q_2^*(-)\\otimes\\mathcal{O}_Z(E)$. \n\nNow we write down the exact sequence defining the cotangent bundle on $S^{[2]}$. Putting left non-zero arrow of the sequence \\eqref{pullbackCotangent} together with the conormal exact sequence of the embedding $Z\\hookrightarrow S\\times S^{[2]}$ twisted by $E$ into a commutative diagram\n\\begin{equation}\n\\label{cotangentHilb2}\n\\begin{tikzcd}\n& 0 \\arrow[d] \\arrow[r] & q_2^*\\Omega_{S^{[2]}}(E) \\arrow[r, \"\\sim\"] \\arrow[d] & q_2^*\\Omega_{S^{[2]}}(E) \\arrow[d]\\\\\n0 \\arrow[r] & \\mathcal{N}^\\vee_{Z\/S\\times S^{[2]}}(E) \\arrow[r]    & q_1^*\\Omega_S(E)\\oplus q_2^*\\Omega_{S^{[2]}}(E)   \\arrow[r]      & \\Omega_Z(E)\n\\end{tikzcd}\n\\end{equation}\nand applying the snake lemma we obtain the exact sequence\n\\begin{equation}\n\\label{snake}\n0 \\longrightarrow \\mathcal{N}^\\vee_{Z\/S\\times S^{[2]}}(E) \\longrightarrow q_1^*\\Omega_S(E)\\longrightarrow j_*\\mathcal{O}_E\\longrightarrow 0.\n\\end{equation}\nAfter pushing forward \\eqref{snake} along $q_2$ we obtain the exact sequence\n\\begin{equation}\\label{cotangent}\n0\\to\\Omega_{S^{[2]}}\\to\\Omega_S^{[2]}\\otimes L\\to\\iota_*\\mathcal{O}_D\\to0.\n\\end{equation}\n\\section{Computation for the Hilberst square of K3 surface}\nFrom now on we assume that $S$ is a K3 surface. We fix some isomorphism $T_S\\stackrel{\\simeq}\\longrightarrow\\Omega_S$. The isomorphism $\\omega_S\\cong\\mathcal{O}_S$ yields $\\omega_Z\\cong\\mathcal{O}_Z(E)$. From the Euler sequence \\eqref{eul} it follows that $\\Omega_{E\/S}\\cong\\mathcal{O}_E(-2)$. From stability of $\\Omega_S$ we have that $\\Hom(\\Omega_S, \\Omega_S)\\cong\\mathbb{C}$. The latter implies that $H^0(S, \\Sym^2(\\Omega_S)) = 0$. Also, we will use the equality $H^0(S, \\Omega_S) = 0$ which by \\cite[Remark 3.19]{krug} and by stability of $\\Omega_S$ implies that $\\Hom(\\Omega_S^{[2]}, \\Omega_S^{[2]})\\cong\\mathbb{C}$.\n\nTo prove the Theorem \\ref{main} in this case it is enough to show the following two equalities\n\\begin{equation}\n\\label{firstVanishing}\n\\Ext^1(\\Omega_S^{[2]}\\otimes L, \\Omega_{S^{[2]}})=0,\n\\end{equation}\n\\begin{equation}\n\\label{secondVanishing}\n\\Ext^2(\\iota_*\\mathcal{O}_D, \\Omega_{S^{[2]}})=0.\n\\end{equation}\n\n\\begin{lemma}\\label{usefulEqualities} The following equalities hold\n\\begin{enumerate}[(i)]\n  \\item $\\Hom(q_1^*\\Omega_S(E), j_*\\mathcal{O}_E) \\cong \\Hom(q_1^*\\Omega_S(E), q_1^*\\Omega_S(E)|_E) \\cong \\Hom(q_1^*\\Omega_S(E)|_E, j_*\\mathcal{O}_E) \\cong\\mathbb{C},$\n  \\item $\\Ext^1(\\iota_*\\mathcal{O}_D, \\iota_*\\mathcal{O}_D) = 0,$\n  \\item $\\Hom(\\Omega_S^{[2]}\\otimes L, \\iota_*\\mathcal{O}_D)\\cong \\mathbb{C},$\n  \\item $\\Ext^2(\\mathcal{O}_{2E}, q_1^*\\Omega_S) = 0,$\n  \\item $\\Ext^k(q_2^*\\Omega_S^{[2]}, q_1^*\\Omega_S(-E))=0$ for $k=0,1$,\n  \\item $\\Ext^k(q_2^*\\Omega_S^{[2]}, j_*\\Omega_{E\/S}(-E))=0$ for $k=0,1$,\n  \\item $\\Ext^2(\\mathcal{O}_{2E}, j_*\\Omega_{E\/S}) = 0$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nAll listed equalities are straightforward consequences of standard adjunctions, properties of the blow-up and projective bundle map $E\\to S$, so we only give sketched proofs.\n\n\\noindent (i) We have\n\\begin{align*}\\Hom(q_1^*\\Omega_S(E), j_*\\mathcal{O}_E)&\\cong\\Hom(p_1^*\\Omega_S, \\sigma_*j_*\\mathcal{O}_E(-E))\\\\\n&\\cong\\Hom(p_1^*\\Omega_S, \\Delta_*\\pi_*\\mathcal{O}_E(1))\\\\\n&\\cong\\Hom(p_1^*\\Omega_S, \\Delta_*T_S)\\\\\n&\\cong\\Hom(\\Omega_S, T_S)\\\\\n&\\cong\\mathbb{C}.\n\\end{align*}\n\nBy the projection formula we have that $\\Hom(q_1^*\\Omega_S(E), q_1^*\\Omega_S(E)|_E)\\cong\\Hom(\\Omega_S, \\Omega_S)\\cong\\mathbb{C}$. Finally, $\\Hom(\\pi^*\\Omega_S(E), \\mathcal{O}_E)\\cong\\Hom(\\Omega_S, T_S)\\cong\\mathbb{C}$. From the fact that $q_1\\circ j = \\pi$ and since the functor $j_*$ is fully faithful on the level of abelian categories, we get $\\Hom(q_1^*\\Omega_S(E)|_E, j_*\\mathcal{O}_E)\\cong\\mathbb{C}$.\n\n\\noindent (ii) Using that $\\cL\\!j^*\\mathcal{O}_{2E} \\cong \\mathcal{O}_E\\oplus\\mathcal{O}_E(-2E)[1]$ and $\\cR\\!\\pi_*\\mathcal{O}_E(-2)\\cong\\omega^\\vee_S[-1]$, by the adjunction and the projection formula we have\n\\begin{align*}\\Ext^k(\\iota_*\\mathcal{O}_D,\\iota_*\\mathcal{O}_D)&\\cong\\Ext^k(\\iota_*\\mathcal{O}_D, q_{2*}j_*\\mathcal{O}_E)\\\\\n&\\cong\\Ext^k(q_2^*\\iota_*\\mathcal{O}_D, j_*\\mathcal{O}_E)\\\\\n&\\cong\\Ext^k(\\mathcal{O}_{2E}, j_*\\mathcal{O}_E)\\\\\n&\\cong\\Ext^k(\\mathcal{O}_E, \\mathcal{O}_E)\\oplus\\Ext^{k-1}(\\mathcal{O}_E, \\mathcal{O}_E(2E))\\\\\n&\\cong H^k(S,\\mathcal{O}_S)\\oplus H^{k-2}(S, \\omega^\\vee_S).\n\\end{align*}\nHence $\\Ext^1(\\iota_*\\mathcal{O}_D,\\iota_*\\mathcal{O}_D) = 0$.\n\n\\noindent (iii) Since $\\Hom(q_1^*\\Omega_S, j_*\\mathcal{O}_E)= 0$ and $\\Hom(q_1'^*\\Omega_S, j_*\\mathcal{O}_E(-E))\\cong\\Hom(\\Omega_S, T_S)\\cong\\mathbb{C}$, from the exact sequence \\eqref{incidence} with $F=\\Omega_S$ we have that $\\Hom(q_2^*\\Omega_S^{[2]}, j_*\\mathcal{O}_E(-E))\\cong\\mathbb{C}$. Then $\\Hom(\\Omega_S^{[2]}\\otimes L, \\iota_*\\mathcal{O}_D)\\cong\\Hom(q_2^*\\Omega_S^{[2]}, j_*\\mathcal{O}_E(-E))\\cong\\mathbb{C}$. \n\n\\noindent (iv) Applying $\\sigma_*$ to the exact sequence\n\\begin{equation}\n\\label{exactSequenceO2E}\n0\\to j_*\\mathcal{O}_E\\to\\mathcal{O}_{2E}(E)\\to j_*\\mathcal{O}_{E}(E)\\to0\n\\end{equation}\nand using the equality $\\cR\\!\\pi_*\\mathcal{O}_E(-1)=0$ we obtain that $\\cR\\!\\sigma_*\\mathcal{O}_{2E}(E)\\cong\\mathcal{O}_\\Delta$. Thus $\\Ext^2(q_1^*\\Omega_S, \\mathcal{O}_{2E}(E))\\cong H^2(S, T_S) = 0$. The assertion then follows from the Serre duality.\n\n\\noindent (v) Applying adjunctions $q_1^*\\dashv q_{1*}$ and $q_{2*}\\dashv q_2^!$ we obtain \n$\\Ext^k(q_2^*\\Omega_S^{[2]}, q_1^*\\Omega_S(-E))\\cong\\Ext^k(q_1^*\\Omega_S, q_2^*\\Omega_S^{[2]})\n$.\nSince $\\cR\\!q_{1*}q_1'^{*}\\Omega_S\\cong H^1(S, \\Omega_S)\\otimes\\mathcal{O}_S[-1]$ we have that $\\Ext^k(q_1^*\\Omega_S,q_1'^*\\Omega_S)=0$ for $k=0,1$. From the exact sequence\n\\[0\\to I_\\Delta\\to\\mathcal{O}_{S\\times S}\\to\\mathcal{O}_\\Delta\\to0\\]\nand the condition $H^1(S, \\mathcal{O}_S) = 0$ we obtain $\\cR\\!p_{1*}I_\\Delta=\\mathcal{O}_S[-2]$. This implies that $\\Ext^k(q_1^*\\Omega_S,q_1^*\\Omega_S(-E))\\cong\\Ext^k(\\Omega_S,\\Omega_S\\otimes\\cR\\!p_{1*}I_\\Delta)=0$ for $k = 0,1$. Now, applying $\\Hom(q_1^*\\Omega_S, -)$ to the incidence exact sequence \\eqref{incidence} with $F = \\Omega_S$, we obtain the desired statement.\n\n\\noindent (vi) Applying $\\sigma_*$ to the sequence \\eqref{exactSequenceO2E} twisted by $E$, we obtain the isomorphism $\\cR\\!\\sigma_*\\mathcal{O}_{2E}(2E)\\cong\\mathcal{O}_\\Delta[-1]$. Together with the isomorphism $\\Omega_{E\/S}\\cong\\mathcal{O}_E(-2)$ it gives\n\\[\\Ext^k(q_2^*\\Omega_S^{[2]}, j_*\\Omega_{E\/S}(-E))\\cong\\Ext^k(\\Omega_S^{[2]}, \\iota_*\\mathcal{O}_D\\otimes L)\\]\n\\[\\cong\\Ext^k(q_1^*\\Omega_S, \\mathcal{O}_{2E}(2E))\\cong\\Ext^{k - 1}(\\Omega_S, \\mathcal{O}_S) = 0\\]\nfor $k = 0, 1$.\n\n\\noindent (vii) We have that $\\Ext^2(\\mathcal{O}_{2E}, j_*\\Omega_{E\/S})\\cong H^1(S, \\omega_S^\\vee)\\oplus H^0(S, \\Sym^2(T_S)) = 0.$\n\\end{proof}\nFrom Lemma \\ref{usefulEqualities}(1) we have that the map $q_1^*\\Omega_S(E)\\to j_*\\mathcal{O}_E$ in the exact sequence \\eqref{snake} factors as the composition of natural maps\n\\begin{equation}\\label{composition}\nq_1^*\\Omega_S(E)\\longrightarrow q_1^*\\Omega_S(E)|_E\\longrightarrow j_*\\mathcal{O}_E,\n\\end{equation}\nwhere the second map is the pushforward along $j$ of the quotient map $\\pi^*\\Omega_S(-1)\\to\\mathcal{O}_E$ in the Euler exact sequence. \n\nConsider the maps\n\\begin{equation}\\label{firstMap}\n\\alpha_k:\\Ext^k(\\Omega_S^{[2]}, \\Omega_S^{[2]})\\longrightarrow\\Ext^k(\\Omega_S^{[2]}\\otimes L, \\iota_*\\mathcal{O}_D), \\,\\,\\, k = 0, 1,\n\\end{equation}\n\\begin{equation}\\label{secondMap}\n\\beta_2:\\Ext^2(\\iota_*\\mathcal{O}_D, \\Omega_S^{[2]}\\otimes L)\\longrightarrow\\Ext^2(\\iota_*\\mathcal{O}_D,\\iota_*\\mathcal{O}_D),\n\\end{equation}\ncoming from the exact sequence \\eqref{cotangent}. By the adjunction $q_2^*\\dashv q_{2*}$ and factorization \\eqref{composition} the map $\\alpha_k$ can be written as the composition\n\\begin{equation}\\label{extifact}\n\\Ext^k(q_2^*\\Omega_S^{[2]}, q_1^*\\Omega_S)\\to\\Ext^k(q_2^*\\Omega_S^{[2]}, q_1^*\\Omega_S|_E)\\to\\Ext^k(q_2^*\\Omega_S^{[2]}, j_*\\mathcal{O}_E(-E)).\n\\end{equation}\nFrom assertions (v) and (vi) of Lemma \\ref{usefulEqualities} it follows that both maps in \\eqref{extifact} are injective, thus $\\alpha_0$ and $\\alpha_1$ are injective as well. Moreover, by Lemma \\ref{usefulEqualities}(iii) we get that $\\alpha_0$ is an isomorphism since it is a map between one-dimensional vector spaces. This implies the equality $\\eqref{firstVanishing}$. \n\nSimilarly, we now decompose $\\beta_2$ as\n\\begin{equation}\\label{ext2factor}\n\\Ext^2(\\mathcal{O}_{2E}, q_1^*\\Omega_S(E))\\to\\Ext^2(\\mathcal{O}_{2E}, q_1^*\\Omega_S(E)|_E)\\to\\Ext^2(\\mathcal{O}_{2E}, j_*\\mathcal{O}_E).\n\\end{equation}\nLemma \\ref{usefulEqualities}(iv) implies the injectivity of the first map in \\eqref{ext2factor}. The injectivity of the second map follows from Lemma \\ref{usefulEqualities}(vii). This shows that $\\beta_2$ is injective, which together with Lemma \\ref{usefulEqualities}(ii) gives the vanishing \\eqref{secondVanishing}. \n\n\\section{General case}\nLet $X$ be an irreducible holomorphic symplectic manifold and $\\mathcal{H} = (I, J, K)$ be the corresponding hyperk\\\"ahler structure. For any triple $a,b,c\\in\\mathbb{R}$ such that $a^2 + b^2 + c^2 = 1$ the operator $L := aI + bJ + cK$ defines a complex structure on $X$. Such a complex structure $L$ is called \\emph{induced by the hyperk\\\"ahler structure}. The space $Q_{\\mathcal{H}}$ of all induced complex structures of $\\mathcal{H}$ is isomorphic to $\\mathbb{C}P^1$ and is called \\emph{the twistor line} of $\\mathcal{H}$. Denote by $\\mathsf{Comp}_X$ the coarse moduli space of complex structures on $X$. Then for each hyperk\\\"ahler structure we have an embedding $Q_{\\mathcal{H}}\\subset\\mathsf{Comp}_X$.\n\\begin{definition}\n\\emph{A twistor path} in $\\mathsf{Comp}_X$ is a collection of consecutively intersecting twistor lines $Q_0,...,Q_n\\subset\\mathsf{Comp}_X$. Two points $I, I'\\in\\mathsf{Comp}_X$ are called \\emph{equivalent} if there exists a twistor path $\\gamma = Q_0,...,Q_n$ such that $I\\in Q_0$ and $I'\\in Q_n$. The path $\\gamma$ is then called \\emph{a connecting path} of $I$ and $I'$.\n\\end{definition}\n\n\\begin{theorem}\\cite[Theorem 3.2]{verb2}\\label{connectinPath}\nAny two points $I, I'\\in\\mathsf{Comp}_X$ are equivalent. \n\\end{theorem}\nNow we recall the definition of a hyperholomorphic bundle over $X$. \n\\begin{definition}\nLet $F$ be a holomorphic vector bundle over $(X, L)$ with a Hermitian connection $\\nabla$ on $F$. The connection $\\nabla$ is called \\emph{compatible with a holomorphic structure} if $\\nabla_v(\\xi) = 0$ for any holomorphic section $\\xi\\in F$ and any antiholomorphic tangent vector $v$. If there exists a holomorphic structure compatible with the given Hermitian connection $\\nabla$, then this connection is called\n\\emph{integrable}. The connection $\\nabla$ is called \\emph{hyperholomorphic} if it is integrable for any complex structure induced by the hyperk\\\"ahler structure. Then $F$ is called a \\emph{hyperholomorphic bundle}.\n\\end{definition}\nFor an induced complex structure $L$ denote by $H^*_L(X, F)$ the holomorphic cohomologies of $F$ with respect to $L$. We mention the following important property of hyperholomorphic bundles.\n\\begin{theorem}\\cite[Corollary 8.1]{verb}\n\\label{dimensionCohomology}\nLet $F$ be a hyperholomorphic vector bundle. Then for any $i\\geqslant0$ the dimension of the space $H^i_L(X, \\mathcal{E}nd(F))$ is independent of an induced complex structure $L$.\n\\end{theorem}\n\nNote that the tangent bundle $T_X$ equipped with the Levi-Civita connection is always hyperholomorphic (see \\cite[Example 2.9(i)]{verb3}). By Theorem \\ref{connectinPath}, for any deformation $X' = (X, I')$, $I'\\in\\mathsf{Comp}_X$ of $(X, I)$ there exists a twistor path $\\gamma$ connecting $I'$ and $I$. Since $T_X$ is hyperholomorphic, the dimension of the cohomology space $H^1(X, \\mathcal{E}nd(T_X))$ is constant along $\\gamma$ by Theorem \\ref{dimensionCohomology}. In the case when $X$ is a manifold of K3$^{[2]}$-type this dimension is equal to zero by the result of Section 3. This proves Theorem \\ref{main}.\n\\section{References}\n\\renewcommand\\refname{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}