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the event - by - event fluctuations of the transverse emission pattern of hadrons in high - energy collisions of identical heavy nuclei have recently attracted much interest experimentally @xcite and theoretically @xcite . when averaged over collision events , the azimuthal angular distribution of emitted hadrons around the beam axis is symmetric with respect to the plane perpendicular to the impact parameter vector @xmath0 between the two nuclei [ au+au or cu+cu at the relativistic heavy ion collider ( rhic ) or pb+pb at the large hadron collider ( lhc ) ] . the event averaged angular distribution @xmath1 where @xmath2 is the angle between @xmath3 and @xmath0 , is therefore completely characterized by the even fourier coefficients @xmath4 . the dominant coefficient , @xmath5 , is called elliptic flow .    owing to quantum fluctuations in the density distributions of the colliding nuclei and finite particle number effects on the distribution of emitted particles , the left - right symmetry is broken in individual collision events . the angular distribution can then be written in the form @xmath6 where @xmath7 describes the tilt angle of the `` event plane '' for each fourier coefficient with respect to the reaction plane defined by the vector @xmath0 . for even @xmath8 , @xmath7 is peaked around zero ; for odd @xmath8 , @xmath7 is randomly distributed . the dominant odd coefficient , @xmath9 , is known as triangular flow . the event averages of the amplitude of the coefficients @xmath4 are found to be constant over a rather large pseudorapidity range ( @xmath10 ) in pb+pb collisions at the lhc @xcite , indicating an approximately boost invariant origin . the main dynamical source of event - by - event fluctuations in the coefficients @xmath4 are believed to be nearly boost invariant fluctuations in the transverse distribution of the energy density at the beginning of the hydrodynamic expansion of the quark - gluon plasma formed in the nuclear collisions . the geometric anisotropy of these fluctuations then translates in an anisotropic transverse collective flow pattern , which manifests itself in anisotropic particle emission . in the color glass condensate model of energy deposition there are two obvious sources of fluctuations in the deposited energy density . one is geometric fluctuations of the position of nucleons in the colliding nuclei at the moment of impact , leading to transverse fluctuations in the density of field generating color charges . this mechanism has been studied widely and is usually described geometrically by the monte - carlo glauber model @xcite . the transverse correlation length of the fluctuations generated by this mechanism will be of the order of the nucleon radius . the other source of energy density fluctuations are fluctuations in the color field strength for a given density of color charges . this mechanism has not been investigated quantitatively up to now . the transverse correlation length generated by color field fluctuations will be dictated by the single scale governing the physics of the color glass condensate , the saturation scale @xmath11 . since @xmath12 is much smaller than the nucleon radius , the color field fluctuations can be expected to govern the microscopic structure of the transverse energy density fluctuations , which then is modulated on longer transverse scales by fluctuations in the nucleon density in the colliding nuclei .    here we calculate the transverse correlation function of the deposited energy density in nuclear collisions in the framework of the gaussian approximation to the color glass condensate originally proposed by mclerran and venugopalan @xcite . in section ii we derive the two - point correlator of the energy density following the collision of two color glass condensates . in section iii we evaluate the resulting integrals and present numerical results for the correlation function for a representative choice of parameters . in the gaussian random source approximation to the color glass condensate model of small@xmath13 gluon structure of atomic nuclei @xcite the probability distribution of color charge density @xmath14 in the transverse plane is assumed to be of the form @xmath15 = \exp\left ( - \frac{1}{g^2\mu^2 } \int d^2x\ , \rho^a({\bf x})^2 \right ) .\ ] ] here @xmath16 represents the area density of color charges in the colliding nuclei , and @xmath17 is called the saturation scale , because it represents the scale at which the small@xmath13 evolution of the gluon density becomes nonlinear due to saturation effects @xcite . owing to the independent contributions of several nucleons to the color field , the gaussian approximation is expected to provide a good description to the color source distribution in colliding nuclei at small @xmath13 @xcite . in the light - cone gauge , the gaussian color charge distribution translates into a gaussian distribution of transverse gauge field strengths . here we will follow the work of lappi @xcite .    to calculate the initial state density fluctuations @xmath18 where @xmath19 denote vectors in the transverse plane , we start from the expression for the deposited energy density of the gauge field given in eq.(10 ) in ref . @xcite : @xmath20 with transverse vector indices @xmath21 . following the collision , the field strength tensor in the region between the receding nuclei only receives contributions from the mixed terms , as the color field of each individual nucleus is a pure gauge and the field strength tensor of each individual nucleus is thus zero outside the nuclear volume : @xmath22 here `` 1 '' and `` 2 '' denote the gauge fields carried by nucleus 1 and 2 , respectively . the field correlator in the color glass condensate model is given by    @xmath23 ~\frac{p_ip_j}{{\bf p}^2 } ~g(|{\bf p}| ) \end{aligned}\ ] ]    where @xmath24 is the fourier transform of the function @xmath25 ~\theta(1-\lambda|{\bf x}| ) \label{eq : g(x)}\ ] ] with the ir cut - off parameter @xmath26 . it is convenient to decompose the momentum quadrupole tensor as follows : @xmath27 where @xmath28 are the familiar pauli matrices . we thus obtain @xmath29\cos[p_2 ( x_2-y_2 ) ] ~\delta_{ij } ~g(|{\bf p}| ) \nonumber \\ & &   + \cos[p_1 ( x_1-y_1)]\cos[p_2 ( x_2-y_2 ) ] ~\sigma^3_{ij}~\frac{p_1 ^ 2-p_2 ^ 2}{{\bf p}^2}~g(|{\bf p}| ) \nonumber \\ & &   - \sin[p_1 ( x_1-y_1)]\sin[p_2 ( x_2-y_2 ) ] ~ \sigma^1_{ij } ~\frac{p_1p_2}{{\bf p}^2}~g(|{\bf p}| ) \big ) \nonumber\\ & = & \frac{1}{2 } \ , \delta_{mn}\delta_{ab}~ \big(\delta_{ij } d({\bf x}-{\bf y } )   + \sigma^3_{ij } e({\bf x}-{\bf y } ) - \sigma^1_{ij } f({\bf x}-{\bf y})\big )   \equiv \delta_{mn}\delta_{ab}~ s_{ij}({\bf x}-{\bf y } ) .\end{aligned}\ ] ] for later use , we will note the values of the individual correlation functions @xmath30 at the origin : @xmath31 the expression for @xmath32 diverges logarithmically for the function @xmath33 given in ( [ eq : g(x ) ] ) , if the gauge coupling @xmath34 is taken as a constant . however , as pointed out by kovchegov and weigert @xcite , the infrared divergence can be removed by including effects from the running of the coupling constant by means of the substitution @xmath35 in the exponent of ( [ eq : g(x ) ] ) . the specific structure of this substitution , sometimes called the `` triumvirate '' structure of the running coupling , is motivated by the form of next - to - leading order corrections to the small@xmath13 evolution of the bfkl kernel in the color dipole approach to parton saturation @xcite . we first evaluate the expectation value of the deposited energy density : @xmath36 recovering the result given in eq . ( 14 ) of ref .  @xcite . next we evaluate the two - point correlator of the energy density : @xmath37 we again make use of the fact that only correlators among fields in the same nucleus are non - zero , which allows us to suppress the labels 1 and 2 : @xmath38 in the spirit of the gaussian approximation we now factorize the correlators of four gauge fields into products of correlators among two gauge fields , ) , because the connection between @xmath39 and @xmath40 is nonlinear @xcite . for a gaussian source charge distribution , these nonlinearities will thus lead to corrections to the energy density correlator obtained here . ] e.  g. : @xmath41    to proceed further we use the symmetry with respect to the color indices @xmath42 and @xmath43 to combine , e.  g. , the second and third term in the large brackets of ( [ eq : epseps ] ) : @xmath44 the second factor is easily shown to vanish : @xmath45    the same holds true for the forth and seventh term , @xmath46 and @xmath47 . after considerable algebra , the fifth , sixth , eighth and ninth terms combine to @xmath48   \left[a_n^f({\bf y } ) a_m^{e'}({\bf y})-a_m^f({\bf y } ) a_n^{e'}({\bf y})\right ]   \big\rangle \nonumber \\ & = & \frac{g^4}{4}f_{abc}f_{a'b'c } f_{efd}f_{e'f'd } \ ,   \big(\delta_{ae}\delta_{a'f'}s_{im}s_{jn } + \delta_{af'}\delta_{a'e}s_{in}s_{jm}\big ) \nonumber \\ & & \times \big(\delta_{bf}\delta_{b'e'}2[s_{in}s_{jm}-s_{im}s_{jn } ]   + \delta_{be'}\delta_{b'f}2[s_{im}s_{jn}-s_{in}s_{jm}]\big ) \nonumber \\ & = & \frac{g^4}{16}n^2(n^2 - 1 ) \left [ d({\bf x}-{\bf y})^2 + e({\bf x}-{\bf y})^2 + f({\bf x}-{\bf y})^2 \right]^2 , \end{aligned}\ ] ]    where we made use of the relation @xmath49 which follows from the jacobi identity . finally we evaluate the first term :    @xmath50 \nonumber \\ & + & \frac{3g^4}{8}n^2(n^2 - 1 )   \left [ d({\bf x}-{\bf y})^2 + e({\bf x}-{\bf y})^2 + f({\bf x}-{\bf y})^2 \right]^2 .\end{aligned}\ ] ]    combining these equations we finally obtain @xmath51 \nonumber \\ & + & \frac{7g^4}{16}n^2(n^2 - 1 )   \left [ d({\bf x}-{\bf y})^2 + e({\bf x}-{\bf y})^2 + f({\bf x}-{\bf y})^2 \right]^2 .\end{aligned}\ ] ] since the functions @xmath30 always appear in the same combination , it makes sense to introduce the abbreviation @xmath52 in terms of which the average deposited energy density and its fluctuation can be expressed as : @xmath53 .\end{aligned}\ ] ] next we simplify the integrals @xmath54 , @xmath55 and @xmath56 . we abbreviate @xmath57 and @xmath58 . we begin with @xmath59 . @xmath60 + \cos[p(z_1\cos \phi -z_2\sin \phi)]\big )   \end{aligned}\ ] ] we substitute @xmath61 in the last term and introduce the notation @xmath62 , @xmath63 : @xmath64   \nonumber \\ & = &   \int_0^{\infty } \frac{p\ , dp}{2\pi}\ , g(p ) ~j_0(pz )   =   \int_0^{\infty } \frac{d^2p}{4\pi^2}\ , g(p ) ~e^{{\rm i}{\bf p}\cdot { \bf z } } \equiv g(z ) , \end{aligned}\ ] ] where we have used eq .  ( 3.715.18 ) from @xcite . similarly we obtain @xmath65   + \cos[p(z_1\cos \phi -z_2\sin \phi ) ] \big)\cos(2\phi )   .\end{aligned}\ ] ] using the same substitutions we find : @xmath66 \cos(2\phi ) \nonumber \\ & = & -\cos(2\psi)\int_0^{\infty } \frac{p\ , dp}{2\pi}\ , g(p ) j_2(pz ) \end{aligned}\ ] ] where we used eqs .  ( 3.715.18 ) and ( 3.715.7 ) from @xcite . finally , a similar calculation yields : @xmath67    we conclude that the function @xmath68 only depends on the distance @xmath69 between the points @xmath70 and @xmath71 .    we can express @xmath72 and @xmath73 in terms of the following integrals : @xmath74 for @xmath75 , namely : @xmath76 @xmath77 we rewrite the integrals as follows : @xmath78 we have to evaluate integrals of the type @xmath79 this integral can be evaluated for @xmath80 using formula ( 6.633.2 ) in @xcite : @xmath81 we are interested in the limit @xmath82 , which means that we can apply the limit of @xmath83 for large arguments : @xmath84 this yields @xmath85    for @xmath86 we use the recursion relation for bessel functions : @xmath87 and apply formula ( 6.512.3 ) from @xcite : @xmath88 with the convention @xmath89 . this implies : @xmath90 when we insert these results into the desired integrals , we find : @xmath91 for the selected parameters . ]     for the selected parameters . ] we now evaluate the average energy density and its fluctuations for a choice of the parameters that is motivated by the initial conditions at which thermal qcd matter is formed in heavy ion collisions at rhic and lhc : @xmath92 note that the result is independent of the value of @xmath26 . the functions @xmath93 and @xmath94 are shown in figs . [ fig : c0 ] and [ fig : c2 ] for these parameter values .    for these parameters , the initial value of the deposited energy density is @xmath95 . this very large energy density quickly decreases due to the longitudinal expansion and reaches much smaller values by the time of thermalization . what matters for us is not the absolute value of the initial energy density , but the relative size and spatial correlation of its fluctuations , @xmath96 . this function is shown in fig . [ fig : deps ] for the parameters listed above .     for the selected parameters . ] as the figure shows , the fluctuations of the initial energy density are locally of similar magnitude as the energy density itself and fall over distances of the inverse saturation scale , here assumed as @xmath97 . this result is in accord with the intuitive picture of the field configuration immediately after the collision in the color glass condensate model , as a bundle of longitudinally stretching random color flux tubes with characteristic transverse width @xmath98 . we have calculated the initial energy density fluctuations in high - energy heavy ion collisions within the gaussian color glass condensate model . these turn out to be very large with a transverse profile determined by the saturation scale @xmath11 . a finite result is only obtained when the `` triumvirate '' running coupling is used , giving additional support to the correctness of eq . 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we calculate the transverse correlation of fluctuations of the deposited energy density in nuclear collisions in the framework of the gaussian color glass condensate model .