modern low frequency (  mhz ) radio telescopes consist of dozens to thousands of low gain antennas , the outputs of which are typically digitized so beamforming and imaging may be performed in the digital domain . in these telescopes ,each individual antenna is integrated with a front end , which typically performs amplification , filtering , impedance transformation , and differential to single - ended ( i.e. balanced to unbalanced ) conversion .the front end must be sensitive to astrophysical signals , but must also be sufficiently linear to be robust to external interference . thus there exists a trade off between sensitivity and linearity .in this paper we present front end design methodologies which consider sensitivity and linearity jointly .system - level sensitivity requirements place well - known constraints on the receiver noise temperature , leading to typical values in the range of a few hundred kelvin  .linearity requirements are less well established .table [ tab : feecomp ] compares the front ends employed in some modern low frequency radio telescopes : the first station of the long wavelength array ( lwa1 )  , the murchison widefield array ( mwa )  , the low frequency array ( lofar ) low - band  , the expanded very large array ( evla ) `` 4-band '' system  , and the eight - meter wavelength transient array ( eta )  . to quantify the sensitivity - linearity trade off, we performed a survey of commercially - available monolithic microwave integrated circuit ( mmic ) amplifiers that might be considered for use in radio telescope front ends . from this survey we identified two amplifiers : the mini - circuits ( mc ) pga-103 , a standout in , and the mc hela-10 , a standout in linearity .we designed amplifiers using pga-103s in ( 1 ) a balanced configuration and in ( 2 ) parallel configuration of balanced amplifiers in order to improve the linearity of the single - ended pga-103 .a front end using two hela-10 amplifiers with input filtering was designed to demonstrate a front end with the highest reported linearity in a modern radio telescope .the relevant performance metrics for these new front end designs are also included in table [ tab : feecomp ] .the remainder of this paper is organized as follows .section [ sec : noisetempreqs ] summarizes the sensitivity requirements for front ends in modern low frequency radio telescopes . in section [ sec : survey ] we present a survey of commercially - available mmic amplifiers .section [ sec : ampmodel ] quantifies the performance of amplifiers consisting of a parallel configuration of balanced amplifiers , in terms of the linearity - sensitivity trade off .sections [ sec : pga ] and [ sec : hela ] present the pga-103 ( high sensitivity ) and hela-10 ( high linearity ) front ends , respectively .section [ sec : demo ] presents the results of a field test demonstration using the front ends presented in section [ sec : designexamples ] .our conclusions are summarized in section [ sec : conc ] .l r r r r l front end & & & & & + & & & & + evla `` 4-band'' & 50 - 86 & 710 & at 74 mhz & 2000 ma @ 17.5 v & ( harden , p. ( nrao ) , personal comm . ) + eta & 27 - 49 & 300 & at 38 mhz & 160 ma @ 12  v & + lwa1 & 10 - 88 & 255 & at 74 mhz & 260 ma @ 15  v & + lofar low - band & 10 - 80 & 180 & at 100 mhz & 120 ma @ 3  v & + mwa & 80 - 300 & 21 & at 100 mhz & 120 ma @ 3  v & +  + hela-10 & 30 - 80 & 700 &  at 50 mhz & 800 ma @ 18 v & section [ sec : hela ] + pga-103 - parallel & 30 - 80 & 131 & at 50 mhz & 90 ma @ 12  v & section [ sec : pga ] + pga-103 - balanced & 30 - 80 & 112 & at 50 mhz & 180 ma @ 12  v & section [ sec : pga ] + pga-103 - single - ended & 30 - 80 & 51 & at 50 mhz & 360 ma @ 12  v & section [ sec : pga ] +as in any other frequency regime , the sensitivity of a low frequency radio telescope is determined primarily by collecting area and noise . however the extraordinarily high sky brightness temperature at low frequencies leads to somewhat different considerations in design criteria and subsequent characterization of instruments .  a model for characterizing the sensitivity - limiting noise delivered to the digitizer in a low frequency instrumentis shown in fig .[ lfa_fnoisemodel ] . for simplicitythis particular model assumes that each element is individually digitized , but the principles remain relevant when analog combining is employed .noise captured by the antenna is conveniently characterized in terms of antenna temperature . for the purposes of this paper defined as the power spectral density ( psd ; e.g. , w hz ) delivered to a load which is conjugate - matched to the antenna impedance , divided by boltzmann s constant  j / k .  as will be explained later , antennas used in low frequency arrays are often poorly - matched to receivers .the effect of impedance mismatch may be taken into account by defining a two - port circuit , identified as an `` antenna interface '' ( ai ) in fig .[ lfa_fnoisemodel ] , which connects the antenna to the rest of the receiver .the ai is assumed to be perfectly - matched to the receiver over the bandwidth of interest , and exhibits input impedance which may or may not vary significantly with frequency . in implementation terms , the ai may be any combination of impedance transforming devices , baluns , and filters ; or the ai may simply a `` null '' two - port representing direct connection between the antenna and receiver . in any event , the ai may be characterized in terms of transducer power gain , which in the present problem may be defined as the ratio of the psd delivered to the receiver , to the psd `` available '' from the antenna . neglecting for the moment any noise that might originate from the ai ( i.e. assuming the ai is lossless ) , where as may be verified using elementary circuit analysis . alternatively be expressed in terms of the `` power wave reflection coefficient '' ) is invalid when the voltage reflection coefficient ( no tilde ) is used in lieu of : i.e. , unless either or is purely real valued . ] .the psd subsequently delivered to the digitizer is where is the gain of the receiver and the second term accounts for the noise contribution from the receiver , expressed as the input - referred equivalent noise temperature .front ends in low frequency radio telescopes are not typically noise - matched to the antenna .antenna impedance in these instruments varies dramatically , so the front end input impedance is typically not equal to the value that would yield optimal noise figure . as a consequence , the minimum possible is not obtained .however , as long as the realized is sufficiently below , improved noise matching will have only a minor impact on sensitivity ( more on this below ; see paragraph including eq .[ eqn : stopcrit ] ) .thus , eq . ( [ eqn : sr ] ) is exact only if is determined for the _ in situ _ input impedance , as opposed to using the value associated with the vendor - specified impedance ; but the effect on the results presented here is expected to be small since the mismatch with the antenna impedance remains large . is comprised of naturally - occurring contributions from the sky and from the ground , plus noise associated with interference . interfering noise originates from human activity and from natural events including lightning and solar bursts  .in contrast to deliberate man - made signals , man - made noise below 300 mhz is normally negligible compared to naturally - occurring noise in the rural locations where radio telescopes are typically built .naturally - occurring interference is intermittent and negligible most of the time , so the resulting antenna temperature can be expressed as where is ground loss efficiency .this efficiency accounts for near - field coupling between the antenna and the ground that results in loss in the ground manifesting as loss in the antenna itself .typical values of range from about ( typical at 38 mhz above untreated earth without a conducting ground screen ) to very close to 1 ( typical above 100 mhz or with a sufficiently large ground screen , as in the demonstration in section [ sec : demo ] ) .the value of is significantly affected by the ground screen . for a perfectly - conducting infinite ground screen , the apparent brightness of the ground is about equal to the apparent brightness of the sky , due to reflection . for no ground screen  k , i.e. ,  k corresponding to the physical temperature of the ground divided by 2 since roughly half of the antenna pattern intersects the ground . for the small ( $ ] in area ) ground screens typically used in modern instruments ,the actual value of is somewhere between these two extremes ( for example , we obtain a value of 390 k at 45 mhz for the field experiment conducted in section [ sec : demo ] ) . in this frequency regime is dominated by the very bright galactic synchrotron background .the brightness temperature of this background is somewhat brighter in the galactic plane and somewhat dimmer in the galactic polar regions , however the total contribution to the antenna temperature of a low - gain antenna isolated from ground loss is well - approximated as where is frequency .( this expression is obtained by a fit to the model described in appendix i of . )this is shown in fig .[ lfa_ftant ] .note that a diurnal variation of about % is present in this contribution , with the maximum corresponding to the transit of the galactic center .n latitude . ]  combining results , the psd delivered to the digitizer is to convey a sense of magnitudes , fig .[ lfa_fnoiseterms ] shows these contributions individually and together for a simple dipole 2-m long by 2 cm in diameter .the antenna is half - wavelength resonant at  mhz . in this result, is obtained using an equivalent circuit model  , , is from eq .( [ eqn : lfa_etsky ] ) ,  k , , and  k. the contributions are shown in the form of input - referred equivalent noise temperature , which is essentially eq .( [ lfa_esr ] ) divided by so that the sum may be interpreted as system temperature ; i.e.   [ cols="^,^ " , ]   figure [ fig : jai_pga_mod ] shows the measured gain and noise temperature for each pga-103 front end .as expected , gain decreases slightly for with respect to , and with respect to , due to losses in the hybrids . increases for the same reason , as does the power consumption , from 1.1 w for the front end to 2.2 w and 4.3 w for the and front ends , respectively .figure [ fig : model ] summarizes the test results at 50 mhz in the form of the analysis of section [ sec : survey ] . for comparison ,theoretical results are shown assuming both the nominal ( datasheet ) and measured amplifier performance , following the analysis of the previous section and accounting for the hybrids loss and phase errors .the measured of the design was within 15 k of the expected value , and the measured iip3 was within 1 db of the expected value .errors can reasonably be attributed to a combination of component manufacturing variances and limitations of the test methods and equipment .the design improves iip3 by 3.8 db and degrades by 61 k ; this compared to 3.6 db and 50 k , respectively , predicted by the analysis using datasheet values from the device .the design improves iip3 by an additional 5 db and degrades by an additional 19 k ; this compared to predicted values of 3.2 db and 16 k , respectively , again using datasheet values for the pga-103 and sbtcj-1w .these results are consistent with the analysis presented in section [ sec : ampmodel ] .  for the amplifier , iip2was measured to be  dbm ( no corresponding value is provided by the manufacturer ) .iip2 for the device is predicted to increase by almost 60 db relative to the device , to nearly  dbm .the achieved value was beyond our ability to measure ( i.e. greater than  dbm ) .the mc hela-10 is the most linear mmic amplifier identified in the survey described in section [ sec : survey ] . here, we describe a new hela-10 based front end for use between 30 - 80 mhz at sites at where increased linearity is required to accommodate strong radio frequency interference ( rfi ) .a block diagram and schematic of the front end are shown in figs . [ fig : fee_bd ] and [ fig : fee_sch ] , respectively .the two hela-10 amplifiers are used as a pre - amplifier and a line driver , respectively .preceding the first amplifier are notch filters designed to trap the strong signals from citizen band ( cb ) and fm radio at 26.527.5 and 88108 mhz , respectively .an absorptive second - order butterworth filter sets the bandpass of the front end to 30 - 80 mhz .the signal path is differential from the antenna through the output of the front end , at which point a balun converts the signal path to single - ended .the front end draws .8 a of dc current at a dc voltage ranging from 16 - 20 v. additionally , the front end implements a three - state calibration system similar to that used by the large aperture experiment to detect the dark ages ( leda )  and experiment to detect the global eor step ( edges )  to facilitate absolute measurement of . the calibration state is determined by the dc voltage used to bias the front end  .figure [ fig : fee_built ] shows a completed front end , affixed to its enclosure , which doubles as a heat sink .figure [ fig : labcommisioning_jai_mod ] shows the typical ( average from the four constructed units ) front end gain and noise temperature .the measured and iip3 are  dbm and .9 dbm , respectively , at 50 mhz , and the achieved iip2 was again beyond our ability to measure .the goal of the demonstration is to accurately measure the galactic noise background .equation [ eqn : lfa_etsky ] approximates the contribution of this background noise to the antenna temperature . a block diagram of the demonstration setup is shown in fig .[ fig : fieldtest ] .the antenna used in this demonstration is a 3.048 m long , 2.5 cm diameter , straight copper dipole positioned parallel to and 1.524 m above a 7.6 m.6 m ground screen , which sits on flat , grass - covered ground .the coaxial cable is 152 m of lmr-400 .the analog receiver provides a gain of about 75 db , 15 - 73 mhz bandwidth ( 3 db ) , and input - referred noise temperature of  k. the coaxial cable , analog receiver , and spectrum analyzer are calibrated out and do not affect the results .  .]  all measurements were taken in glen alton , va ; a rural area about 65 km northwest of blacksburg , va .all measurements were taken at approximately the same local sidereal time , within one hour .figure [ fig : jai_skyspectra_s_r ] shows the measured psd for each front end .the expected hf - band rfi is present below 30 mhz , as well as the fm band between 87 - 108 mhz .the large signal between 60 - 66 mhz is a digital television ( dtv ) station ( channel 3 ) .the total power available from the antenna ( i.e. , applied to the fee input ) in the spectrum shown is approximately  dbm .however the signal available from the antenna is strongly dominated by local am stations transmitting at the 5 - 10 kw level , resulting in an actual total power of at least dbm from our antenna . the power associated with the am stationswas not accurately measured at the time of the experiment and may have actually been significantly greater .the antenna temperature for each front end was estimated by solving eq .( [ eqn : sr ] ) for as follows : for the pga-103 front ends , and where measured in the lab . for the hela-10 front end , and where determined using the internal calibration system .the single - ended pga-103 front ends were connected to the antenna through an mc adt1 - 1 balun transformer , which has a 1:1 impedance ratio and a measured insertion loss of about 0.5 db over the considered frequency range .the antenna impedance was estimated using ansys hfss em modeling software , and fig .[ fig : jai_skyspectra_g_t ] shows the resulting .the antenna temperature is related to the temperature distribution surrounding the antenna by  \ ] ] where and are the antenna s normalized power pattern and beam solid angle , respectively , and is the element of solid angle .the contributions of and to are the upper and lower hemisphere portions of the integral : where we have assumed ground loss to be negligible , such that and the ground perfectly reflects the sky temperature distribution . using a moment method model ( nec2 , assuming a ground conductivity of 5 ms / m and dielectric constant of 15 ) to calculate and assuming a uniform brightness temperature distribution with a constant temperature given by eq .( [ eqn : lfa_etsky ] ) , we find across the entire frequency range of interest .thus , the sky temperature was estimated by .   of the antenna interface for the field measurements in sec .[ sec : demo ] . ]figure [ fig : jai_skyspectra_cal ] shows estimated using eqs .( [ eqn : ta ] ) and ( [ eqn : t_sky_meas ] ) and the data collected from each front end , and the modeled from eq .( [ eqn : lfa_etsky ] ) .the hela-10 front end is not noticeably affected by the interference , due in part to the highly linear hela-10 amplifiers and in part to the cb , fm , and bandpass filters .remaining disagreement with the model is attributed to uncertainty in the antenna impedance .although the pga-103 front end appears to be experiencing severe compression , we are not certain if this is classical gain compression , or whether it is a more complex non - linear behavior arising from the spectral distribution of the interference ( i.e. , the am - dominated antenna output power ) or some other effect resulting from the power level handled by a single pga-103 in this case .the improved linearity of the balanced ( ) and parallel ( ) pga-103 front is apparent , as the estimated approaches the model for increasing . without the interference , or with the addition of input filters similar to those on the hela-10 front end, we expect the pga-103 front ends would yield a similar to the result from the hela-10 front end .   using the pga-103 front ends .( _ solid : _ ) estimated using the hela-10 front end .( _ dotted : _ ) model of given in eq .( [ eqn : lfa_etsky ] ) . ]this paper has considered the design of front ends for modern low frequency radio telescopes with respect to the trade off between sensitivity and linearity .a survey of commercially - available mmic amplifiers was performed and an informal analysis was used to identify the state - of - the - art .two outlying amplifiers , which outperformed the majority of comparable amplifiers , were selected for demonstration .a high sensitivity front end using the pga-103 amplifier was presented with  k , however this amplifier was significantly affected by interference in a field demonstration .a high - linearity front end using hela-10 amplifiers and cb / fm - band filters was less sensitive than the pga-103 front end (  k ) , but was robust to the interference and facilitated a greatly improved detection of the galactic synchrotron background .balanced and parallel configurations of amplifiers were also considered with respect to the linearity - sensitivity trade off .front ends employing pga-103 amplifiers in balanced and parallel configurations reduced the effects of non - linearity , as expected from theory , at a cost to both sensitivity and power consumption .this work was supported in part by the virginia tech bradley foundation and in part by the national science foundation through grants ast-1105949 and ast-1139963 .the authors acknowledge the assistance of x. gomez of virginia tech , for providing the hfss antenna impedance model and for assisting with the field measurement .       \\     & \quad + a_2 \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [ \cos \left ( ( \omega_1 - \omega_2 ) t \right ) + \cos \left ( ( \omega_2 + \omega_1 ) t \right ) \right ] \\     &\quad + \frac{3}{4 } a_3 \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t \right ) + \cos \left ( ( 2\omega_1 + \omega_2 ) t \right ) \right . \\     & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_2 - \omega_1 ) t \right ) + \cos \left ( ( 2 \omega_2 + \omega_1 ) t \right ) \right ] \end{split } % \begin{split } %  y^{a}(t ) & = a_1 \frac{1}{\sqrt{c } } a \left [ \cos \left ( \omega_1 t \right ) + \cos \left ( \omega_2 t \right ) \right ] \\%    & \quad + a_2 \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [ \cos \left ( ( \omega_1 - \omega_2 ) t \right )  \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( \omega_2 + \omega_1 ) t \right ) \right ] \\%    & \quad + \frac{3}{4 } a_3 \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t \right ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_1 + \omega_2 ) t \right ) \right. \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_2 - \omega_1 ) t \right ) \right .\\  %    & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2 \omega_2 + \omega_1 ) t \right ) \right ] % \end{split}\ ] ]  and \\     & \quad + a_2 \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [ \cos \left ( ( \omega_1 - \omega_2 ) t \right ) + \cos \left ( ( \omega_2 + \omega_1 ) t  - 2\phi - 2\delta\right ) \right ] \\     &\quad + \frac{3}{4 } a_3 \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t - \phi - \delta \right ) + \cos \left ( ( 2\omega_1 + \omega_2 ) t - \phi - \delta \right ) \right . \\     & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_2 - \omega_1 ) t - \phi - \delta \right ) + \cos \left ( ( 2 \omega_2 + \omega_1 ) t - \phi - \delta \right ) \right ] \end{split }  % \begin{split } %  y^{(b)}(t ) & = a_1 \frac{1}{\sqrt{c } } a \left [ \cos \left ( \omega_1 t - \phi - \delta \right ) + \cos \left ( \omega_2 t - \phi - \delta \right ) \right ] \\%    & \quad + a_2 \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [ \cos \left ( ( \omega_1 - \omega_2 ) t \right ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( \omega_2 + \omega_1 ) t  - 2\phi - 2\delta\right ) \right ] \\ %    & \quad + \frac{3}{4 } a_3 \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t - \phi - \delta \right ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_1 + \omega_2 ) t - \phi - \delta \right ) \right . \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left. + \cos \left ( ( 2\omega_2 - \omega_1 ) t - \phi - \delta \right ) \right. \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2 \omega_2 + \omega_1 ) t - \phi - \delta \right ) \right ] % \end{split}\ ] ]  respectively , where the dc , harmonic distortion , and gain compression terms have been suppressed ( e.g. assuming filtering ) for the purposes of determining iip2 and iip3 .the output hybrid modifies each term in by a phase shift of and adds it to , such that \\     & \quad + a_2 \frac{1}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [ \cos \left ( ( \omega_1 - \omega_2 ) t - \phi - \delta \right ) + \cos \left ( ( \omega_2 + \omega_1 ) t - \phi - \delta \right ) \right . \\     & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( \omega_1 - \omega_2 ) t \right ) + \cos \left ( ( \omega_2 + \omega_1 ) t  - 2\phi - 2\delta\right ) \right ] \\     &\quad + \frac{3}{4 } a_3 \frac{2}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t - \phi - \delta \right ) + \cos \left ( ( 2\omega_1 + \omega_2 ) t - \phi - \delta \right ) \right . \\     & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_2 - \omega_1 ) t - \phi - \delta \right ) + \cos \left ( ( 2 \omega_2 + \omega_1 ) t - \phi - \delta \right ) \right ] \end{split } % \begin{split } %  y(t )  & = a_1 \frac{2}{c } a \left [ \cos \left ( \omega_1 t - \phi - \delta \right ) + \cos \left ( \omega_2 t - \phi - \delta \right ) \right ] \\ %    & \quad + a_2 \frac{1}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [ \cos \left ( ( \omega_1 - \omega_2 ) t - \phi - \delta \right ) \right . \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~ + \cos \left ( ( \omega_2 + \omega_1 ) t - \phi - \delta \right )  \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~ + \cos \left ( ( \omega_1 - \omega_2 ) t \right )  \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~ \left. + \cos \left ( ( \omega_2 + \omega_1 ) t  - 2\phi - 2\delta\right ) \right ] \\%    & \quad + \frac{3}{4 } a_3 \frac{2}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t - \phi - \delta \right ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_1 + \omega_2 ) t - \phi - \delta \right ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_2 - \omega_1 ) t - \phi - \delta \right ) \right. \\  %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2 \omega_2 + \omega_1 ) t - \phi - \delta \right ) \right ] % \end{split}\ ] ] for a hybrid , , and for commercially - available hybrids operating at vhf and uhf is typically less than . expanding the trigonometric functions in order to isolate ,applying small angle approximations for , and keeping only the first - order terms , the output becomes \\     & \quad - \delta a_2 \frac{1}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [  \sin \left ( ( \omega_1 - \omega_2 ) t  \right ) - \sin \left ( ( \omega_2 + \omega_1 ) t \right ) \right ] \\     & \quad - \frac{3}{4 } a_3 \frac{2}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t \right ) + \cos \left ( ( 2\omega_1 + \omega_2 ) t \right ) \right . \\     & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_2 - \omega_1 ) t \right ) \right .+ \cos \left ( ( 2 \omega_2 + \omega_1 ) t \right ) \\     & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \delta \sin \left ( ( 2\omega_1 - \omega_2 ) t \right ) + \delta \sin \left ( ( 2\omega_1 + \omega_2 ) t \right )  \\     & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \delta \sin \left ( ( 2\omega_2 - \omega_1 ) t \right ) \left .+ \delta \sin \left ( ( 2 \omega_2 + \omega_1 ) t \right ) \right ] \end{split } % \begin{split } %  y(t )  & \cong - a_1 \frac{2}{c } a \left [ \cos(\omega_1 t )  + \cos ( \omega_2 t ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~ \left .+ \delta \sin(\omega_1 t )  + \delta \sin ( \omega_2 t ) \right ] \\%    & \quad - \delta a_2 \frac{1}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^2 \left [  \sin \left ( ( \omega_1 - \omega_2 ) t  \right ) \right . \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .- \sin \left ( ( \omega_2 + \omega_1 ) t \right ) \right ] \\%    & \quad - \frac{3}{4 } a_3 \frac{2}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^3 \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t \right ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_1 + \omega_2 ) t \right ) \right .\\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \cos \left ( ( 2\omega_2 - \omega_1 ) t \right ) \right . \\  %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \cos \left ( ( 2 \omega_2 + \omega_1 ) t \right ) \\%    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \delta \sin \left ( ( 2\omega_1 - \omega_2 ) t \right ) \\%    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \delta \sin \left ( ( 2\omega_1 + \omega_2 ) t \right )  \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \delta \sin \left ( ( 2\omega_2 - \omega_1 ) t \right ) \\ %    & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left .+ \delta \sin \left ( ( 2 \omega_2 + \omega_1 ) t \right ) \right ] % \end{split}\ ] ] of which are the linear , fundamental tones , are the im2 products , and \quad , \quad i , j=1,2 , j \neq i\ ] ] are the im3 products .to determine iip2 and iip3 , we need the root - mean - square ( rms ) values of the fundamental tones and the intermodulation products .the mean is taken by averaging over each tone s period ( denoted by ) , and any terms containing are considered negligible . the result is    \bigg)^2 dt \bigg]^{1/2 } \\%    & % \begin{split } %  \sqrt { \left\langle y_{im2}^2(t)\right\rangle } & = \bigg [ \frac{1}{t } \int_0^t \bigg ( \delta a_2 \frac{1}{\sqrt{c } } \left(\frac{1}{\sqrt{c } } a \right)^2  \\ %  & \quad ~~~~~~~~ \times \left [  \sin \left ( ( \omega_2 + \omega_1 ) t \right ) \right ] \bigg)^2 dt \bigg]^{1/2 } \\%    & = \frac{1}{\sqrt{2 } } \delta a_2 \frac{1}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^2  % \end{split}\ ] ]   \bigg)^2 dt \bigg]^{1/2 } \\%  & % \begin{split } %  \sqrt { \left\langle y_{im3}^2(t)\right\rangle } & = \bigg [ \frac{1}{t } \int_0^t \bigg ( - \frac{3}{4 } a_3 \frac{2}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^3  \\ %  & \quad ~~~~~~~~ \times \left [ \cos \left ( ( 2\omega_1 - \omega_2 ) t \right ) \right . \\ %  & \quad ~~~~~~~~~~~  \left .+ \delta \sin \left ( 2(\omega_1 - \omega_2 ) t \right ) \right ] \bigg)^2 dt \bigg]^{1/2 } \\%  & \cong \frac{3}{4 \sqrt{2 } } a_3 \frac{2}{\sqrt{c } } \left ( \frac{1}{\sqrt{c } } a \right)^3 % \end{split}\ ] ]

Abstract: radio telescope front ends must have simultaneously low noise and sufficiently - high linearity to accommodate interfering signals . typically these are opposing design goals . for modern radio telescopes operating in the hf ( 330 mhz ) and vhf - low ( 3088 mhz ) bands , the problem is more nuanced in that front end noise temperature may be a relatively small component of the system temperature , and increased linearity may be required due to the particular interference problems associated with this spectrum . in this paper we present an analysis of the sensitivity - linearity trade off at these frequencies , applicable to existing commercially - available monolithic microwave integrated circuit ( mmic ) amplifiers in single - ended , differential , and parallelized configurations . this analysis and associated findings should be useful in the design and upgrade of front ends for low frequency radio telescopes . the analysis is demonstrated explicitly for one of the better - performing amplifiers encountered in this study , the mini - circuits pga-103 , and is confirmed by hardware measurements . we also present a design based on the mini - circuits hela-10 amplifier , which is better - suited for applications where linearity is a primary concern .   = 1  ; ; ;


originally formulated in 1970 , the belinskii - khalatnikov - lifshitz ( bkl ) conjecture states that as one approaches a space - like singularity , ` terms containing time derivatives in einstein s equations dominate over those containing spatial derivatives ' .this implies that einstein s partial differential equations are well approximated by ordinary differential equations ( odes ) , whence the dynamics of general relativity effectively become local and oscillatory .the time evolution of fields at each spatial point is well approximated by that in homogeneous cosmologies , classified by bianchi .the simplest of these are the bianchi i metrics which have no spatial curvature and the bianchi ii metrics which have ` minimal ' spatial curvature . according to the bkl conjecture ,the dynamics of each spatial point follow the ` mixmaster ' behavior a sequence of bianchi i solutions bridged by bianchi ii transitions .finally , with the significant exception of a scalar field , matter contributions become negligible to quote wheeler , matter does nt matter " .  in the beginning, the conjecture seemed to be coordinate dependent and rather implausible .however , subsequent analysis by a large number of authors has shown that it can be made precise and by now there is an impressive body of numerical and analytical evidence in its support .it is fair to say that we are still quite far from a proof of the conjecture in the full theory .but there has been outstanding progress in simpler models .in particular , berger , garfinkle , moncrief , isenberg , weaver and others showed that , in a class of models , as the singularity is approached the solutions to the full einstein field equations approach the ` velocity term dominated ' ( vtd ) ones obtained by neglecting spatial derivatives . andersson andrendall showed that for gravity coupled to a massless scalar field or a stiff fluid , for every solution to the vtd equations there exists a solution to the full field equations that converges to the vtd solution as the singularity is approached , _ even in the absence of symmetries_. these results were generalized to also include -form gauge fields in . in these vtd modelsthe dynamics are simpler , allowing a precise statement of the conjecture that could be proven . in the general case ,the strongest evidence to date comes from numerical evolutions .berger and moncrief began a program to analyze generic cosmological singularities . while the initial work focused on symmetry reduced cases , more recently garfinkle performed numerical evolution of space - times with no symmetries in which , again , the mixmaster behavior is apparent .finally , additional support for the conjecture has come from a numerical study of the behavior of test fields near the singularity of a schwarzschild black hole .  with growing evidence for the bkl conjecture ,it is natural to consider its implications to quantum gravity .the conjecture predicts a dramatic simplification of general relativity near space - like singularities , which are precisely the places where quantum gravity effects are expected to dominate .a promising approach to analyze this issue is provided by loop quantum cosmology ( lqc ) where there are now several indications that the quantum gravity effects become important only when curvature or matter density are about a percent of the planck scale .therefore it is quite possible that , generically , spatial derivatives become negligible compared to the time derivatives already when the universe is sufficiently classical . in this case a quantization of the effective theory with odes , that descends from techniques applicable in the full theory , could provide a reliable qualitative picture of quantum gravity effects near generic space - like singularities .if , on the other hand , the bkl behavior sets in only in the planck regime , this strategy would not be viable .but since there is no reason to trust einstein s equations in this regime , then the conjecture would also not have a physically interesting domain of validity .lqc is the result of application of the principles of loop quantum gravity ( lqg ) to symmetry reduced cosmological models .initial study of the k=0 friedmann lemaitre robertson walker ( flrw ) models revealed that the quantum geometry effects underlying lqg provide a natural mechanism for the resolution of the big bang singularity .subsequent more complete analysis led to a detailed understanding of the physics in the planck regime and also showed that although these effects are very strong there capable of replacing the big bang with a quantum bounce they die extremely rapidly so as to recover general relativity as soon as the curvature falls below planck scale .these results were then extended to include spatial curvature in and a cosmological constant in .more recent investigations reveal that if matter satisfies a non - dissipative equation of state , lqc resolves all strong curvature singularities of the flrw models , including , e.g. , those of the ` big - rip ' or ` sudden death ' type .also , it is now known in lqc that the bianchi i and ii and ix singularities are resolved .  in view of the bkl conjecture ,these results , together with further support from the ` hybrid ' quantization of gowdy models , suggest that there may well be a general theorem to the effect that all space - like singularities of the classical theory are naturally resolved in lqg .however , it is difficult to test this idea using the current formulations of the bkl conjecture since these approaches are motivated by the theory of partial differential equations rather than by hamiltonian or quantum considerations ( see e.g. , .in particular , most approaches perform a rescaling of their dynamical variables by dividing by the trace of the extrinsic curvature .it is difficult to promote the resulting variables to operators on the lqc hilbert space . in the analysis presented here, we reformulate the bkl conjecture in a way better suited to lqc and explore the resulting system both analytically and numerically.  in lqg one begins with a first order formalism where the basic canonical variables are a density - weighted triad and a spin - connection .in section [ s2 ] we will begin by recalling this hamiltonian formulation of general relativity . in section [ s3 ]we rewrite this theory using a set of variables that are motivated by the bkl conjecture .rather surprisingly , the core of this theory can be formulated using ( density weighted ) fields with only internal indices ; _ space - time tensors never feature _ ! to understand the implications of the bkl conjecture to lqg , we need to express the conjecture using this hamiltonian framework .this task is carried out in section [ s4 ] .we provide a weak and a strong version of the conjecture .the key idea is to say that , as one approaches space - like singularities , the exact system is well approximated by a truncated system which features only time derivatives .non - triviality of the formulation lies in the choice of variables and specification of how limits are taken .our procedure satisfies a number of stringent requirements .in particular , one can either first truncate the hamiltonian and then obtain the equations of motion or first obtain the full equations of motion and then truncate them ; the two procedures commute . in section [ s5 ]we study the truncated hamiltonian system and explore its dynamics in some detail .we show that it exhibits all the known features such as the ` u - map ' and spikes .thus , _ the hamiltonian framework we were led to by lqg considerations successfully captures the mixmaster dynamics faithfully ._ therefore , in addition to providing a viable point of departure to analyzing the fate of generic space - like singularities in lqg , it should also be useful in analytical and numerical investigations of the bkl conjecture in classical general relativity itself . in section [ s6 ]we summarize the main results and comment on their relation to those of other works .the two appendices contain more technical material .appendix [ a1 ] introduces densities in a coordinate - free manner .this notion is important because the basic variables in our formulation of the bkl conjectures are scalar densities of weight 1 . in the main text , for simplicitywe have set the shift and the lagrange multiplier of the gauss constraint equal to zero .appendix [ a2 ] contains the full equations without these restrictions .we will consider space - times of the form where is a compact , oriented 3-dimensional manifold ( without boundary ) .we will formulate general relativity in terms of first order variables , the point of departure of lqg .these consist of pairs of fields consisting of a ( density weighted ) orthonormal triad , and its conjugate momentum which on solutions will correspond to extrinsic curvature .the fundamental poisson bracket is given by \{ ^a_i(x ) , k_b^j(y ) } = _i^j ^a_b ^3(x - y ) herein , early letters , denote spatial indices while denote internal indices which take values in the lie algebra of .tildes are used to capture density weights of quantities ; a tilde above indicates that the quantity transforms as a ( tensor ) density of weight 1 and a tilde below will denote a ( tensor ) density of weight .the internal indices can be freely raised and lowered using a fixed kinematical metric on .the phase space spanned by smooth pairs will be denoted by .these phase space variables are related to their arnowitt , deser and misner ( adm ) counterparts by ^a_i ^b_j ^ij= q^ab + k_a^i ^b_i = k_a^b where is the metric on the leaf , its determinant , and the extrinsic curvature of . in terms of these variableswe perform a 3 + 1 decomposition of space - time to obtain as hamiltonian a sum of constraints with lagrange multipliers : h[,k]=_^3m- - n^a _ a + _the lagrange multipliers , , the lapse and shift , are related to the choice of slicing and time in the standard fashion , and is related to rotations in the internal space .phase space functions , , and are the scalar , vector , and gauss constraints ( with density weights respectively ) , given by [ co ] & ( , k ) & - - 2^a_[i ^b_j ] k_a^i k_b^j + & _ a(,k ) & 4 d_[a(k_b]^i ^b_i ) + & ^k(,k ) & _ i^jk ^a_j k_a^i where is the scalar curvature of the metric .the overall sign and numerical factors in the constraints are chosen so they reduce to the standard adm constraints upon solving the gauss constraint . can be written in terms of the triad and its inverse or in terms of the triad and the connection compatible with the triad , which is defined by d_a ^b_i + _ ijk _ a^j ^bk = 0 , _a^j = - _ bk d_a ^b_i ^ijk .( note that acts only on tensor indices ; _ it treats the internal indices as scalars ._ ) although is determined entirely by for now it is convenient to use all three fields , and in our classical analysis : in our formulation of the bkl conjecture and will be the relevant degrees of freedom near the singularity , so it is natural to express the theory in terms of them .the equations of motion are obtained by taking poisson brackets with the hamiltonian on the phase space : [ edot0]^a_i & = & \{^a_i , h[,k ] } + [ kdot0]_a^i & = & \ { k_a^i , h[,k ] } . is the phase space underlying lqg .the basic variables used there are obtained by a simple canonical transformation on  : ( ^a_i , k_a^i ) ( a_a^i,^-1 _ a^i ) a_a^i = _ a^i + k_a^i , being the barbero - immirzi parameter of lqc .( in classical general relativity , space - time equations of motion are independent of the value of this real parameter . ) for simplicity of presentation we will introduce our formulation of the bkl conjecture using although it will be clear that our framework can be readily recast in terms of .in order to formulate the bkl conjecture in this system , one needs to specify two things : what kind of derivatives are to dominate as one approaches the singularity and what kind are to become negligible ? and , what are the quantities whose derivatives are to be treated as negligible ? in this section we first motivate and introduce a set of variables and a derivative operator and then use them to formulate the conjecture .the main idea is as follows .the accumulated evidence to date suggests that the spatial metric becomes degenerate at the space - like singularity whence its determinant vanishes there .( in particular , this is borne out in the numerical simulations of solutions with two commuting killing fields the so - called space - times which include gowdy models . )we will focus on the class of singularities where this occurs . in this caseone would expect that if we rescaled fields which are ordinarily divergent at the singularity with appropriate powers of , the rescaled quantities would have well defined limits .now , the density weighted triad is obtained by rescaling of the orthonormal triad , which is divergent at the singularity , by . in examples ,the factor of not only gives a well defined limit , but the limit in fact vanishes .therefore , contraction by can serve to tame fields which would otherwise have been divergent at the singularity .this consideration leads us to construct scalar densities by contracting with , and . as noted above , since contraction with suppresses the divergence of and , the combination is expected to remain finite at the singularity .let us then set [ cp ] _i^j & : = ^a_i k_a^j - ^a_k k_a^k _ i^j + _ i^j & : = ^a_i _ a^j - ^a_k _ a^k _ i^j . these two fields , and will turn out to be the relevant variables near the singularity in our bkl framework .in particular , we will show below that the constraints of general relativity can be expressed in terms of polynomials of these basic variables and their derivatives .therefore if the basic variables and their derivatives remain finite at the singularity , the constraints will also continue to hold there .since the hamiltonian of the theory is a linear combination of these constraints , dynamics of the basic variables will meaningfully extend to the singularity .  beyond the possibility of being bounded at the singularity ,an important feature of these variables is that they have _ only internal indices _ which can be freely raised and lowered using the fixed , kinematic , internal metric ; the dynamical metric which diverges at singularities is not needed . under diffeomorphisms and as density weighted _ scalars _ on .because of this feature , statements about their asymptotic properties can be formulated much more easily than would be possible if they were tensor fields .( for a coordinate free introduction to densities , see appendix [ a1 ] ) . to illustrate why these variables are likelyto be well defined at the singularity , let us consider the bianchi i model . because of spatial flatness , we can work in an internal gauge in which everywhere .what about and ? in terms of the commonly used proper time ,the metric is given by and the singularity occurs at . since , we have in the bianchi i chart .in addition , due to the second constraint on the exponents , whence the density weighted triad vanishes at the singularity as and is finite there for each .we further introduce a derivative operator defined by the contraction of with : _ i : = ^a_i d_a .the expectation is that this contraction will have the effect of suppressing terms containing as we approach the singularity .thus , will be the spatial derivatives we were seeking which , when acting on certain quantities , will be conjectured to be negligible near the singularity .is non - trivial only on tensor indices ) .however since its action on a function does not yield the exterior derivative , is not a connection . if we were to formally treat as a connection , it would have torsion , which is related to : } f = -\tilde{t}^k_{\;\;ij } \dt_k f ] . inwhat follows , often acts on scalar densities .this action is given explicitly in appendix [ a1 ] . ]the variable is related to the momentum (conjugate to the 3-metric ) in the adm phase space by . encodes information in the spatial derivatives of the triad : [ ga ] ^ij = - _^klj _ k ^a_l .note that , although the depend on spatial derivatives of the triad and are often sub - dominant to , it turns out that they are not always negligible in the approach to the singularity .indeed , this behavior is observed in the truncated system , which is discussed in section [ s4 ] .it is rather than the triads themselves that will feature directly in our formulation of the conjecture .  for simplicity of notation ,_ from now on we will drop the tildes . _thus , from now on each of carries a density weight , while the lapse field carries a density weight .the scalar and the vector constraint functions and ( introduce below ) carry density weight 2 while the gauss constraint carries density weight 1 .  by making use of ( [ cp ] ) and ( [ ga ] ) ,functions of and their covariant derivatives can be rewritten in terms of and their derivatives .the scalar curvature for example can be expressed entirely in terms of and its derivatives : q=-2^ijk _ i ( _ jk ) - 4 _ [ ij ] ^[ij ] - _ ij ^ji + ^2 consequently , the constraints can be re - expressed _ entirely _ in terms of and their derivatives ( with no direct reference to or even the determinant of the 3-metric ) : [ s1 ] s & = & 2^ijk _ i ( _ jk ) + 4 _ [ ij ] ^[ij ] + _ ij ^ji - ^2 + _ij ^ji - ^2 + [ v1 ] v_i & = & - 2_j _i^j - 2 _ jkl _i^j ^kl - _ ijk ^jk + 2_ijk ^jl _l^k + [ g1 ] g^k & = & ^ijk _ji .here we have converted the co - vector index on the vector constraint to an internal index by contracting it with . since the is assumed to be non - degenerate away from the singularity the constraint defines the same constraint surface as the original vector constraint introduced in ( [ co ] ) .notice here that the constraint can be easily decomposed into those terms that contain the derivative and those that do nt .the equations of motion for , , can be written in a similar form .these can be obtained using the full poisson brackets ( [ edot0]),([kdot0 ] ) or by directly computing poisson brackets of and with the scalar / hamiltonian constraint . to streamline the second calculation ,let us specify the poisson brackets between , , and : [ pb1 ] \{e^a_i(x ) , _j^k(y ) } & = & ( e^a_j(x ) _i^k- e^a_i ( x)_j^k ) ( x , y ) + [ pb2 ] \ { _ i^j(x),_k^l(y ) } & = & ( _ k^j(x ) _ i^l - _k^j ) ( x , y ) + [ pb3 ] \ { f_ij ^ij , g_kl ^kl } & = & ( f_ij g_kl ( ^kj ^il + ^jl ^ik ) + ^jlm ^ik g_kl _m f_ij ) + [ pb4 ] \{e^a_i(x ) , ^j_k(y ) } & = & 0 \{_i^j(x),_k^l(y ) } = 0 , where are smooth test scalar fields . the equations of motion obtained by taking poisson brackets with the scalar constraintare then given by [ cdot1]&^ij = - & ^jkl _ k ( n ( _ l^i -_l^i ) ) + n [ 2 ^(i_k ^|k|j ) + 2^[kj ] _ k^i - ^ij ] + [ pdot1]&^ij = & ^jkl _ k ( n ( 1/2 _ l^i -_l^i ) ) - ^klm _ m ( n _ kl ) ^ij+ 2^jkm ^[ik ] _ m ( n ) + & & ( ^i ^j - ^k _ k ^ij ) n + n [ - 2 ^(ik ) _ k^j + ^ij- 2 ^[kl ] _ [ kl]^ij ] and where we have set the shift to zero to reduce clutter .( for non - zero shift , see appendix [ a2 ] . )note that the equation of motion for is a simple ode .note also that , as was the case with constraints , the equations of motion for and can again be written in terms of scalar densities and the derivative _ only_.this motivates us to ask for an evolution equation for the derivative operator .since ignores internal indices , it suffices to consider its action just on scalar densities of weight .we have : [ ddot1 ] _i s_n = [ _ i ( n )]s_n - n _i^j_j s_n . _ thus we have cast all the constraint as well as evolution equations as a closed system involving only _ , , and .these equations can then be used as follows . on an initial slice ,we construct from a pair of canonical variables .then we can deal exclusively with the triplet .the pair satisfies constraints if and only if the triplet satisfies ( [ s1])([g1 ] ) .given such a triplet , we can evolve it using ( [ cdot1]),([pdot1]),([ddot1 ] ) _ without having to refer back to the original canonical pair _ .these two sets of equations have some interesting unforeseen features .first , as already mentioned , the basic triplet has _ only internal indices _ : our basic fields are _ scalars _ on ( with density weight 1 ) .it would be of considerable interest to investigate if this fact provides new insights into the dynamics of 3 + 1 dimensional gravity .second , these equations _ do not refer to the triad _ .suppose we begin at an initial time where is derived from an .then these constraint and evolution equations ensure that _ is derivable from a triad at all times_. furthermore , we can easily construct that triad directly from a solution to these equations : first solve ( [ cdot1])([ddot1 ] ) and then simply integrate the ode [ edot ]^a_i = -n p_i^j e^a_j at the end .third , the structure of the constraint and evolution equations in terms of is remarkably simple since only low order polynomials of these variables are involved .finally , thanks to our rescaling by , our basic triplet ( as well as ) are expected to have a well behaved limit at the singularity .a close examination of our equations shows that they allow the triad become to become degenerate during evolution .so , strictly ( as in lqg ) we have a generalization of einstein s equations .  to summarize, we have found variables which remain finite at the singularity in examples and rewritten einstein s equations as a _ closed system of differential equations _ in terms of them .therefore , this formulation may be useful for proving global existence and uniqueness results and rigorous exploration of fields near space - like singularities . finally ,although for simplicity we have set shift and the smearing field equal to zero , the features we just discussed hold more generally ( see appendix [ a2 ] ) .  to conclude ,let us examine the action of the vector and the gauss constraints on our basic variables .( the action of the scalar constraint yields the evolution equations which we have already discussed . )since the vector constraint generates a combination of spatial diffeomorphisms and internal rotations , it is standard to subtract a multiple of the gauss constraint to define the diffeomorphism constraint : [ diff ] v_i = v_i - 2(_i^j - _ i^j ) g_j we can then smear both constraints to obtain g [ ] = _ ^3 m ^k g_k v[n ] = _ ^3 m n^i v_i where s a scalar with density weight so that is the standard lapse and , as before , has density weight zero .the action of  ] are necessarily negligible under our assumptions. this fact will be useful throughout our analysis .next , note that we have expressed general relativity in the form of a constrained theory in terms of our basic variables , , , and .our constraints are composed of quadratic terms in our basic variables and terms of the form and .we can therefore split each constraint into two parts terms which contain no derivatives and those which do .similarly the equations of motion can be split into terms that contain derivatives and those that do not . with this background ,we can state two versions of our conjecture .  * weak conjecture : as the singularity is approached the terms containing derivatives in the constraints and equations of motion are negligible in comparison to the polynomial terms .thus , as the singularity is approached the constraints and equations of motion approach those found by setting derivative terms to zero .*  we define the truncated theory to be the system defined by setting -derivative terms to zero , [ truncdef]_i _ j^k=_i _ j^k=_i n=_in_j = _ [ ij]=0 , in the equations of motion ( [ cdot1])-([ddot1 ] ) and constraints ( [ s1])-([g1 ] ) . thus , the weak conjecture says that the equations of motion can be well approximated by those of the truncated theory in the vicinity of the singularity .note that this does not imply that the _ solutions _ of the full equations of motion will approach the solutions to the truncated equations as the singularity is approached .this additional condition is captured in the strong version as follows .* strong conjecture : as the singularity is approached the constraints and the equations of motion approach those of the truncated theory and in addition the solutions to the full equations are well approximated by solutions to the truncated equations . *  with the strong conjecture the solution of the full einstein equations will asymptote to solutions of the truncated system defined by ( [ truncdef ] ) . in the followingwe will analyze this truncated system .  not only are the truncated constraints purely algebraic , but they involve only quadratic combinations our basic variables : [ s2 ] s_(t ) & : = & _ ij ^ji - ^2 + _ij ^ji - ^2 + [ v2 ] v_i^(t ) & : = & - _ ijk ^jk + 2_ijk ^jl _ l^k + [ g2 ] g^k_(t ) & : = & ^ijk _ji , the truncated gauss constraint is in fact exact because ( [ g1 ] ) involves no derivative terms , while the scalar and the diffeomorphism constraints are genuinely truncated .the infinitesimal transformations ( [ gt1 ] ) - ( [ vt3 ] ) generated by the full constraints contain derivative terms that are now assumed to be negligible in comparison to the polynomial terms .ignoring the negligible terms leads us to the following transformations on the basic fields : [ gt1 ] \{p_ij , g()}_t & = & 2 _kl(j p_i)^k ^l + [ gt2 ] \{c_kl , g()}_t & = & 2 _kl(j c_i)^k ^l + [ vt1 ] \{p_ij , v(n)}_t & = & 4 _kl(j p_i)^k n^m ( c_m^l - _ m^l ) + [ vt2 ] \{c_ij , v(n)}_t & = & 4 _kl(j c_i)^k n^m ( c_m^l - _ m^l ) + [ st1 ] \{c_ij , s(n)}_t & = & -2n ( 2 c_k(i p^k_j ) - pc_ij ) + [ st2 ] \{p_ij , s(n)}_t & = & -2n ( - 2 c_ik c^k_j + c c_ij ).  the gauss constraint continues to generate internal rotations , but , whereas in the full theory transforms as ( the contraction of the triad with ) a connection , after truncation both and transform as tensors .the vector constraint also generates internal rotations , since the diffeomorphism constraint generates only negligible terms .we arrived at the truncated equations of motion by first obtaining the full equations and then applying the truncation to them i.e. , by setting spatial derivative terms to zero .but we could also have first truncated the constraints to obtain ( [ s2])- ( [ g2 ] ) and then computed their truncated poisson brackets with the basic variables .this leads to a consistency check of our scheme : do the two procedure yield the same ` truncated equations of motion ' in the end ?the answer is in the affirmative . this fact is illustrated by the following ` commutativity diagram ' :    note that the operation of truncation , the final truncated system , and hence the consistency requirement mentioned above depends crucially on one s choice of basic variables and notions of space and time derivatives . for example , if we had adopted the more ` obvious ' strategy and used triads rather than as basic variables , we would have been led to set to zero in the truncation procedure since would then be derived quantities , obtained by taking the derivative of .this truncation would have led us just to bianchi i equations ._ the resulting bkl conjecture would have been manifestly false ._ thus , considerable care is needed to arrive at variables which satisfy a closed set of equations in a hamiltonian framework , suggest a natural way to make the heuristic idea of ignoring spatial derivatives in favor of time derivatives precise , lead to the above commuting diagram , and a version of the bkl conjecture that is compatible with the large body of analytical and numerical results that has accumulated so far .it is rather striking that the variables automatically satisfy these rather stringent criteria .in this section we will analyze the truncated system in some detail and show that its solutions reproduce the expected bkl behavior .the section is divided into three parts . in the first we regard as fields on the full phase space ,obtain the truncated poisson brackets between them and truncated constraints . in the second we solve and gauge fix the vector and the gauss constraints of the truncated theory .the result is a finite dimensional , reduced phase space with a single constraint which is well suited to serve as a starting point for quantization inspired by the bkl conjecture .in the third part we discuss several features of solutions to this hamiltonian theory .in particular , we will find that they exhibit bianchi i phases with bianchi ii transitions .since the truncated equations of motion can be formulated entirely in terms of , let us truncate the poisson brackets ( [ pb2 ] ) , ( [ pb3 ] ) we obtained between them by setting the negligible terms on the right side to zero . since the full poisson bracket ( [ pb3 ] ) involves smearing fields and we first need to specify which terms involving them are to be regarded as negligible .the most natural avenue is to construct to and only from the basic fields ( and their derivatives ) .then the terms containing derivatives of the smearing fields will also be negligible and hence vanish in the truncation .the resulting truncated poisson brackets between and are then given by : [ tpb1 ] \{_i^j(x ) , _k^l(y ) } _ t & = & ( _ k^j _ i^l + ^jl _ik)(x ) ( x , y ) + [ tpb2 ] \ { _ i^j(x ) , _ k^l(y)}_t & = & ( _ k^j _ i^l - _ i^l _ k^j)(x ) ( x , y ) + [ tpb3 ] \{_i^j(x ) , _k^l(y ) } _ t & = & 0 . these poisson brackets suffice to determine the equations of motion because the truncated hamiltonian constraint ( [ s2 ] ) is algebraic in and .they are now odes , [ cdot2]_ij = n [ 2 c_k(i p^k_j ) - p c_ij ] _ ij = n [ - 2 c_ik c^k_j + c c_ij ] , so the truncated dynamics at any one spatial point decouple from those at other points .  this system has some notable features .first , we have a _ closed system _ expressed entirely in terms of and at any fixed point .furthermore , the equations of motion ( [ cdot2 ] ) and constraints ( [ s2])-([g2 ] ) are at most quadratic in these variables . in the full theory, the triad does not appear explicitly in the equations ( [ cdot1 ] ) - ( [ ddot1 ] ) but is implicitly present through . upon truncation , even this implicit dependence disappears .second , as in the full theory , one can first solve the equations of motion for and and then evolve the triad at that point at the end by solving an ode .third , the truncated scalar constraint ( [ s2 ] ) is symmetric under interchange of and and , by adding a multiple of the gauss constraint , the vector constraint can be made anti - symmetric under this interchange : [ vred ] |v^i_(t ) : = ^ijk _ j^l _ kl .however , this symmetry is broken at the level of equations of motion because the truncated poisson algebra does not have a simple transformation property under this interchange .because fields at distinct points decouple , to study the truncated system from the viewpoint of differential equations , one can simply restrict oneself to a single spatial point . however , this is not directly possible in the hamiltonian framework because even in the truncated theory , the poisson brackets ( [ tpb1 ] ) , ( [ tpb2 ] ) involve .but one can introduce a _ subspace _ of the full phase space tailored to our truncation .given a point in consider the pair of density weighted fields it determines .the phase space point will be said to be _ homogeneous _ if there exists an internal gauge and a nowhere vanishing scalar density of weight such that the ( density weight zero ) scalar fields are constants on ( and } = 0 $ ] ) .( fixing a is equivalent to fixing a 3-form on ; see appendix [ a1 ] . ) clearly , the truncated dynamics leaves this _ homogeneous sub - space _ of the phase space invariant .more importantly , _ is invariant under full dynamics _ : if the -derivatives are initially zero they remain zero under the full equations of motion .the hamiltonian dynamics on fully captures the truncated dynamics at any fixed spatial point on .  _remark : _ since the triads in the full phase space have been assumed to be non - degenerate , they are also non - degenerate in .however , as examples suggest , one would expect them to be become degenerate in the limit to the space - like singularity where , however , would continue to be well behaved ( and some of them may even vanish ) .it is therefore of some interest to extend the homogeneous subspace by adding ` limit points ' which have this behavior .this construction is not needed in our analysis . however , since it may be useful in future investigations , we will conclude this subsection with a brief summary .let us allow the density weighted triads to become degenerate such that the subspaces spanned by the non - degenerate directions of vector fields are integrable .( if this condition is satisfied for one nowhere vanishing scalar density , it is satisfied for all . )thus , in the degenerate case we obtain preferred 2 or 1 dimensional sub - manifolds on .we can extend the phase space by including such degenerate if , in addition , the resulting pair is regular , is symmetric and the pair is homogeneous along the preferred lower dimensional sub - manifolds of .key questions for the bkl conjecture are then : i ) does the hamiltonian flow on naturally extend to this extension ? ; and ii ) do generic dynamical trajectories flow to it ?   since is symmetric but is not , the homogeneous subspace is not a symplectic sub - manifold of the full phase space .but it turns out that one can obtain a symplectic manifold by solving and gauge fixing the truncated vector and the gauss constraints .it will be referred to as the _ reduced phase space _ , .the gauss constraint ( [ g2 ] ) is equivalent to asking that is symmetric and then the vector constraint ( [ v2 ] ) is equivalent to asking that as matrices , and should commute . to gaugefix the gauss constraint , we first note the transformation properties ( [ gt1 ] ) and ( [ gt2 ] ) of and under the action of the gauss constraint .it is easy to verify that , because and commute , the requirement that they both be diagonal gauge - fixes the gauss constraint completely .it turns out that the diagonality requirement also fixes the vector constraint .this may seem surprising at first .but note that the combination of the vector and the gauss constraint of eq ( [ vred ] ) again generates internal gauge rotations , where , however the generator is a ` q - number ' , i.e. , depends on the phase space variables : , where is the shift used to smear the vector constraint .the fact that the gauge fixing of the vector constraint does not impose additional requirements on ` cures ' the mismatch in the degrees of freedom in the homogeneous subspace ( arising from the fact that while is symmetric , is not . ) .so far are fields on , each carrying density weight 1 .since these fields are homogeneous , symmetric and diagonal , the reduced phase space is 6 dimensional .it is convenient to coordinatize it with just six numbers , , with : c_1 : = _ ^3 m c_1 ^ 1 ; p^1 : = _ ^3 m p_1 ^ 1 ; where the integrals are well defined because we have completely fixed the internal gauge , in that gauge the integrands are all densities of weight 1 , and is compact . from now on we will focus on the description of in terms of and .  the symplectic structure on is given by the poisson brackets : and , the delta - distributions on the right hand side of truncated poisson brackets ( [ tpb1 ] ) - ( [ tpb2 ] ) on have now disappeared . to write the truncated constraints ( [ s2 ] ) , ( [ v2 ] ) in terms of , one first fixes a nowhere vanishing scalar density of weight 1 ( i.e. , a 3-form ; see appendix a ) .one then multiplies these constraints to obtain constraints with density weight zero . finally , by noting that , etc , where is the volume of with respect , one obtains the equations of motion for given below . ]\{p^i , p^j } = \{c_i , c_j } = 0 \{p^i , c_j } = 2^i_j c_j .the scalar or hamiltonian constraint [ bklham ] c^2 - c_ic^i + p^2 - p_ip^i=0 now generates the equations of motion via poisson brackets : [ kadot ] _i = n c_i ( c- 2 c_i ) + [ gadot ] _ i = -n c_i ( p - 2 p_i ) here and in what follows we use the summation convention also for the indices and have set p = p_1+p_2+p_3 c = c_1+c_2+c_3 . as a side remark , we note that is a fixed point of our system for each , whence the sign of each along any dynamical trajectory is fixed by the initial conditions .therefore , away from the ` planes ' , we can , if we wish , perform a change of variables to and work with the canonically conjugate pair . however , in what follows , we will continue to work with .  finally , recall that in the bkl conjecture ` the only matter that matters ' is a scalar field .let us therefore extend our gravitational reduced phase space to include a massless scalar field .denote the conjugate momentum by so that . then on this extended reduced phase space the hamiltonian constraint is given by [ bklham2 ] c^2 - c_i c^i + p^2 - p_ip^i - = 0 the equations for and are still given by ( [ kadot ] ) and ( [ gadot ] ) while those of the scalar field are simply and .the hamiltonian flow in fully captures the gauge invariant properties of the truncated dynamics of fields at any one fixed spatial point on .let us therefore focus on this hamiltonian system .although the basic constraint and evolution equations on are just odes , they have a rich structure ; indeed they incorporate the dynamics of all bianchi type a models .since the analysis of bianchi ix is already quite complicated and required considerable effort , we will follow the strategy used in and analyze implications of the reduced equations near fixed points .there are two sets of fixed points of the dynamics , i.e. , points at which :   , , , , and   and .the first set of fixed points corresponds essentially to a dimensional reduction of our theory and is therefore highly unstable . to show that our truncation captures the standard features associated with the bkl behavior near singularities, it will suffice to focus on the second set which , we will now show , in fact corresponds to the kasner solutions . one can show that the solutions to the scalar constraint are such that _ all three are positive or all three are negative_. choice of positive signs turns out to be necessary and sufficient for the singularity to appear at as per our previous conventions .let us return for a moment to the homogeneous phase space and set lapse , the fiducial scalar density for which is homogeneous , diagonal , with entries .we can then solve the evolution equation ( [ edot ] ) for the triad in terms of .finally let us set p_i = 1- = e^-pt/2 .then the space - time metric computed from is given by [ kasnermetric ] ds^2 = -d^2 + ^2p_1dx_1 ^ 2 + ^2p_2dx_2 ^ 2 + ^2p_3dx_3 ^ 2 so that the singularity lies at ( or ) . by definition ,the constants satisfy p_1 + p_2 + p_3 = 1and the hamiltonian constraint 2p_i p^i- p^2 + ^2=0 on translates to the familiar quadratic kasner constraint p_1 ^ 2 + p_2 ^ 2 + p_3 ^ 2 = 1-p_^2p_^2 = . for each value of ,these constraints on the define a 1-parameter family of solutions , the intersection of a plane with a 2-sphere .one can check that if all the are positive while if solutions exist only if one of the is negative .we will now show that this distinction plays the key role for the stability of the solution .let us now move away slightly from a kasner fixed point and consider the hamiltonian trajectory through the new point : p_i = p_i + p_i , c_i = c_i + c_i .then , the evolution equations for the perturbations are of the form = ( p^2 ) = -n c_i ( p - 2p_i ) + ( c p ) for definiteness , let us set . then and similarly for .now is positive since all three are positive .therefore , if all are positive ( i.e. if ) , the evolution equation for is of the type , whence the perturbation will decay , implying stability . in terms of the canonical variables describing the scalar field ,this occurs when the scalar field is large : .this stability is in accordance with the andersson - rendall results on approach to space - like singularity in presence of a massless scalar field in full general relativity .   in vacuum , starting from a point near the kasner fixed point surface .initial data is , , , , , ( in blue , in green , in red ) .since none of the initial vanish , as expected from analytical considerations , there is a series of separate taub transitions between kasner states .time has been rescaled by a power of 1/4 to allow multiple transitions to be shown on a single plot . ]   in vacuum , starting from a point near the kasner fixed point surface , with the same initial data as in fig .[ clong ] ( in blue , in green , in red ) .the largest eigenvalue , , transits first . after this transition, becomes the largest eigenvalue , now making unstable . in timeall three tend to zero . in terms of parameters used in the kasner metric ( [ kasnermetric ] ) ,the initially expanding direction starts contracting at the end of the transition and initially contracting starts expanding . ]  let us now consider the complementary case where . by the above reasoning , now is negative for some .for definiteness , let us take to be the largest of the s initially so that is negative which implies that will grow and we have instability . in this case, we can not use perturbative analysis for the pair ; it is necessary to keep all order terms in and . for simplicity ,let us set initially .then values of will not change during evolution and equations for simplify , [ stability ] & = & - n c_1 ^ 2 + & = & - n c_1 ( p_2 + p_3 - p_1 ) = -n c_1 p p_1 , which can be solved exactly to obtain [ cptrans ] p_1(t ) & = & p_2 + p_3 - 2 ( 2 n ( t - t_o ) ) + c_1(t ) & = & 2 sech ( 2 n(t - t_o ) ) . these are the bianchi ii solutions written in our variables . here , the unstable variable , rapidly increases and then decays to zero . during that time the transitions between one kasner solution to another . in the asymptotic limits we have : [ pout ] p_1(- ) & = & p_2 + p_3 + 2 = ( + ) ^2 + p_1(+ ) & = & p_2 + p_3 - 2 = ( -)^2 .(in practice the asymptotic limitsare achieved quickly , thanks to the hyperbolic functions of time . )the result of the transition is that , which was originally was the largest of the three , has transitioned to a lower value . by a change of variables to the used in ( [ kasnermetric ] )it is apparent that the eigenvalue corresponding to the negative exponent is the one which has transitioned , and is positive at the end of the transition .since the singularity lies at , this means that the initially expanding direction now contracts , and one of the two contracting directions now expands . indeed , ( [ pout ] ) is precisely the ` u - map ' in variables .  in this analysiswe have made the simplification that initially .if one starts from a generic point in the vicinity of the kasner fixed point set and still with as the largest of the three initially , there would again be a transition of the type ( [ cptrans ] ) .but as decreases , after a _ finite _ time either or will now be the largest eigenvalue and making the corresponding unstable .that pair will then evolve according to ( [ cptrans ] ) .this general scenario was borne out in a large class of simulations of the reduced equations of motion .figs [ clong ] and [ plong ] illustrate this dynamical behavior for generic initial data near the kasner surface .the taub transitions are easy to see in fig [ plong ] : even though none of the are initially zero , the taub transitions are well described by the analytical expressions ( [ cptrans ] ) .fig [ cgeneric ] illustrates the dynamical behavior in cases where the initial data is quite far from the kasner surface .note that even in this case , the decrease in time so that , although we start far away from the kasner surface , dynamics drives the state to the kasner surface .   in vacuum , starting from a point away from the kasner fixed point surface .initial data are , , , , , ( in blue , in green , in red ) . even though we start out far from the kasner surface ( where all vanish ) , dynamics drives the state to the kasner surface .again , time has been rescaled by a power of 1/4 to allow multiple transitions to be shown on a single plot . ]we can also draw some lessons for the full theory from this behavior of the truncated system . recall that the dynamical trajectories discussed above can be thought of as representing the evolution of fields at a fixed spatial point .let us therefore return to and consider fields .now , generically , we will encounter a point where the all vanish , while being non - zero is the neighborhood of the point . as we noted in section [ s5.2 ] , the sign of is preserved throughout the evolution .this is in particular true during the taub transitions where the magnitude of grows .therefore , on ` one side ' of , a will positive and increasing in magnitude , while on the ` other side ' it will be negative and increasing in magnitude .therefore its derivative will increase rapidly .similarly the under taub transitions the values of will change from those of one kasner solution to another _ except at the point _again , this dynamics will generate a large derivative at .thus , analysis of the reduced system suggests that _ spikes _ will occur in the full system ( [ cdot1 ] ) - ( [ pdot1 ] ) .as is well known , these spikes were found in numerical simulations and , more recently , also in analytical treatments .whenever spikes appear , the key assumption underlying bkl truncation is brought to question because the spatial derivatives are large at the spikes .the key issue for the bkl conjecture and for the application to quantum gravity we proposed in section [ s1] is whether the time derivatives still dominate generically , as they do in examples .let us summarize .using analytical and numerical methods we showed that there exists a well defined subspace of the full phase space which exhibits exactly the properties expected in the bkl conjecture .our procedure to arrive at the reduced system is more direct than those available in the literature . in ,for example , elimination of the off - diagonal components and the anti - symmetric parts of involves an additional assumption , beyond ignoring spatial derivatives in favor of time derivatives : these quantities are identified as part of the ` stable subset ' ( variables that are expected to decay rapidly as the singularity is approached ) and then set to zero to obtain the truncated equations . in our treatment , on the other hand , the fact that the antisymmetric part of is negligible is directly implied by the assumption that the derivatives are negligible and the constraints _ imply _ that the variables and can be simultaneously diagonalized which , furthermore , completely fixes the gauge .thus , our hamiltonian framework naturally led to a diagonal gauge , enabling us to quickly zero - in on the essential variables and eliminating the need to keep track of the dynamics of extraneous variables involving frame rotations .finally , the framework easily led us to the mixmaster behavior a series of bianchi i phases interspersed by bianchi ii transitions .we recovered the ` u - map ' for these transitions , and observed the behavior expected from the andersson and rendall analysis when a scalar field of large enough magnitude is introduced .we began with the hamiltonian formulation of general relativity underlying lqg where the basic fields are spatial triads with density weight 1 , spin connections they determine , and extrinsic curvatures .based on examples that have been studied analytically and numerically , it seems reasonable to expect that the determinant of the spatial metric would vanish and the trace of the extrinsic curvature would diverge at space - like singularities .( this expectation is in particular borne out in the numerical simulations of g2 space - times . )one can therefore hope to obtain quantities which remain well - defined at the singularity either by multiplying the natural geometric fields by suitable powers of or dividing them by suitable powers of . in the commonly used framework due to ugglaet al , one chooses to divide by .one first introduces the so - called hubble normalized triad by rescaling the orthonormal triad by , and then constructs a set of hubble normalized fields by contracting and with .these fields are expected to have regular limit at the space - like singularity .einstein s equations expressed in terms of them naturally suggests a truncation and the truncated system successfully describes the expected oscillatory bkl behavior .the resulting form of the bkl conjecture is supported by numerical evolutions of full general relativity carried out by garfinkle . however , because there is no underlying hamiltonian framework , this approach does not easily lend itself to non - perturbative quantization . even if such a framework were to be constructed , because of the presence of the factor , it would be difficult to introduce quantum operators corresponding to the hubble rescaled fields .motivated by quantum considerations , we adopted the complementary strategy of multiplying geometrical fields by .the lqg hamiltonian formulation we began with already features a density weighted triad with _exactly _ the desired property : . since is expected to vanish at the singularity , one can hope to use in place of the hubble normalized to construct a new set of fields to formulate the bkl conjecture .indeed , ( modulo trace terms ) our basic variables and were obtained simply by contacting the spatial indices of and by .furthermore , because vanishes in the limit , the operator provided a key tool in the formulation of the bkl conjecture : asymptotically , and should become ` negligible ' relative to and .now , in exact general relativity , time derivatives of and can be expressed in terms of their derivatives , purely algebraic ( and at most quadratic ) combinations of and , the lapse and its derivatives ( see ( [ s1])([ddot1 ] ) ) .therefore , if in the limit the derivatives of the basic fields become negligible compared to the fields themselves , we are naturally led to conclude that time derivatives would dominate the spatial derivatives .this chain of argument led to our formulation of the bkl conjecture .this rather simple idea depends on the fact that the structure of einstein s equations has an interesting and unanticipated feature : as we saw in section [ s3 ] , once the triplet is constructed from the triad and the extrinsic curvature on an _ initial slice _ , the constraint and evolution equations can be expressed entirely in terms of the triplet .given a solution to these equations , the spatial triad ( and hence the metric ) can be recovered at the end simply by solving a total differential equation ( [ edot ] ) .this is a surprising and potentially deep property of einstein s equation .it played essential role in our formulation of the bkl conjecture and could well capture the primary reason behind the bkl behavior observed in examples and numerical simulations .since our framework is developed systematically from a hamiltonian theory , its bkl truncation naturally led to a truncated phase space .the specific truncation used has an important property : the truncated constraint and evolution equations on the truncated phase space coincide with the truncation of full equations on the full phase space . on the truncated phase spacewe could solve and gauge - fix the gauss and vector constraints to obtain a simple hamiltonian system ( which encompasses all bianchi type a models ) .solutions to this system were explored both analytically and numerically .we showed that they exhibit the bianchi i behavior , the bianchi ii transitions and spikes as in the analysis of symmetry reduced models and numerical investigations of full general relativity .therefore , as explained in section [ s1 ] , an appropriate quantization of the truncated system , e.g. , a la loop quantum cosmology , could go a long way toward understanding the fate of generic space - like singularities in quantum gravity .  in sections[ s3 ] , [ s5.1 ] and [ s5.2 ] , we restricted ourselves to vacuum equations .the addition of a massless scalar field is straightforward and was carried out in the reduced phase space framework in section [ s5.3 ] . if the energy density in the scalar field is small , one again has bianchi ii transitions and spikes .however , once the energy density exceeds a critical value , these disappear and the asymptotic dynamics at any spatial point is described just by the bianchi i model with a scalar field without transitions .thus , our truncated system faithfully captures the main features generally expected from the analysis of andersson and rendall in full general relativity coupled to a massless scalar field or stiff fluid .thus , although the initial motivation came from quantum considerations , _ our formulation of the bkl conjecture , and the form of the field equations both in the full and truncated versions , should be useful also in the analytical and numerical investigations of singularities in classical general relativity ._  we will conclude with a discussion comparing our approach with that of uggla , ellis , wainwright and elst ( uewe ) ( ) .the hubble normalized variable used in their formulation of field equations are give by [ hn1 ] _ ij & = & 3k^-1 e^a_(i k_|a|j ) - k^-1 e^a_k k_a^k _ij + [ hn2 ] n_ij & = & -3k^-1 e^a_(i _ |a|j ) + 3k^-1 e^a_k _ a^k _ij + [ hn3 ] a_i & = & -_i^jk 3k^-1 e^a_j _ a^k + [ hn4 ] _ i & = & 3k^-1 e^a_i _ a .these variables are especially useful because they are scale invariant : they are unchanged under a constant rescaling of the space - time metric . because of this property and because of the ` regulating ' factor in their expressions , it is hoped that in the limit as one approaches the space - like singularity , these variables will remain finite and their derivative will become negligible .we began with quite a different motivation and our focus was on constructing a hamiltonian framework rather than on differential equations . since our emphasis was on constructing phase space variables that can be readily promoted to well - defined quantum operators , from the start we avoided the use of factors such as .as a result , our basic variables and are _ not _ scale invariant .could we have made a different choice which is also well suited for quantization and at the same time enjoyed scale invariance ?the answer is in the negative for the following reason . under constant conformal rescalings of the space - time metric, we have , , and .now , in the analysis of approach to singularity , scale invariant quantities are directly useful only if they are space _ scalars _ and it is _ not _ possible to construct scale invariant scalars using just sums of products of these fields , i.e. , without introducing fields such as for which it is difficult to construct quantum operators .even if one introduces additional non - dynamical fields , such as fiducial frames to construct scalars , for natural choices of these frames , scale invariant components of fields such as typically diverge at the singularity .thus , with our motivation , it does not seem possible to demand scale invariance of the basic variables that are to feature in the bkl conjecture .our viewpoint is that the most important feature of the hubble normalized variables is that although the orthonormal triad typically diverges as one approaches a space - like singularity , diverges even faster , making the combination go to zero at the singularity .furthermore , it goes to zero at a sufficient rate for its contraction with and in ( [ hn1 ] ) ( [ hn4 ] ) to tame the a priori divergent behavior of these fields . instead of dividing the orthonormal triad by which one expects to diverge at the singularity ,our strategy was to multiply it by the volume element which , in examples , goes to zero at the singularity .this difference persists also in the treatment of the lapse .the uewe framework assumes that the ( scalar ) lapse is such that admits a limit while we assume that the density weighted lapse admits a well - defined limit at the singularity .thus , in both cases , the standard scalar lapse goes to zero so the singularity lies at .the key scale invariant uewe variables which are expected to be well behaved at the singularity are related to our via n_ij = 6 p^-1 c_(ij)&and & _ ij = -6 p^-1 p_(ij ) + 2 _ ij , , + c_(ij ) = - n_ij & and & p_(ij )= ( _ ij - 2 _ ij ) and the two sets of lapse fields are related by : = k n .if one focuses only on the structure of differential equations near space - like singularities , the two reduced systems would in essence be equivalent if admits a finite , nowhere vanishing limit at the singularity .this condition holds for bianchi i models and also bianchi ii which describe the transitions between bianchi i epochs .in fact in the bianchi i model , and our density weighted triad has the _ same _ dependence on proper time as the hubble normalized triad .thus , although the motivations , starting points and procedures used in the two frameworks are quite different , surprisingly , in the end the basic variables and equations are closely related .we would like to thank woei - chet lim , alan rendall , yuxi zheng and especially david garfinkle and claes uggla for discussions .this work was supported in part by the nsf grant phy0854743 and the eberly and frymoyer research funds of penn state .since the basic variables that feature in our formulation of the bkl conjecture are scalars on of density weight 1 , in this appendix we briefly recall a coordinate independent framework for describe densities .the underlying idea is due to wheeler and the detailed framework was developed by geroch ( see , e.g. , ) .this framework goes hand in had with penrose s abstract index notation .because the primary application in this paper is to our fields on we will focus on scalar densities on 3-manifolds .but generalization to tensor densities on n - manifolds is straightforward .fix an oriented 3-manifold and fix a orientation thereon .denote by the space of smooth , positively oriented , no - where vanishing , totally skew tensor fields on . clearly , given any two elements and in , there exists a ( strictly ) positive function such that .this fact will be used repeatedly .  in this paper , a scalar density of weight a map from to the space of ( real valued ) smooth functions on : , such that : s_n(e^ ) = ^n s_n(e ) here can be any real number but in most applications in general relativity it is an integer .( in quantum mechanics , on the other hand , states are ( complex - valued ) densities of weight on the configuration space . )since have density weight , let us make a short detour to discuss the case .fix any 3-form on .it determines a canonical scalar density of weight : s_1(e ) : = s_abc e^abc .conversely , since is a linear mapping from to smooth functions , it determines a canonical 3-form .thus , our basic variables could also be taken to be 3-forms on which take values in second rank tensors in the internal space .the standard adm phase space of general relativity can be similarly coordinatized by positive definite metrics and tensor fields which are symmetric in and totally skew in  .finally note that every metric determines a canonical volume 3-form which has positive orientation and satisfies .therefore it also determines a canonical scalar density of weight , called the _ square root of the determinant _ of : for all .this definition can be extended to density weighted tensor fields in an obvious fashion .note that every carries a natural totally skew tensor density of weight , called the levi - civita density : ^abc ( e ) = e^abc e given any metric on , the square root of its determinant , , can also be expressed as .finally , given a derivative operator on tensor fields on , we can extend its action on densities of weight in a natural manner . is a 1-form with the same density weight , given by ( d_a s_n)(e ) = d_a ( s_n(e ) ) - n _ a s_n(e ) e , where the first term on the right hand side is just the gradient of the function and the 1-form is given by .therefore the action of the derivative operator introduced in the main text is given by : ( d_i s_n)(e ) = d_i(s_n(e ) ) - n ( e^a_i_a ) s_n(e)e .since the derivative operator we considered ignores internal indices , this equation gives the action of on and by regarding these basic fields simply as scalar densities with weight .in the main text we restricted the equations of motion to the case where the shift is zero as is the lagrange multiplier for the gauss constraint . in this appendixwe give the equations of motion in full generality for both full general relativity and in our reduced system .the full equations of motion for and are as follows . & ^ij = - & ^jkl _ k ( n ( _ l^i -_l^i ) ) +n [ 2 ^(i_k ^|k|j ) + 2^[kj ] _ k^i - ^ij ] + & & + n^k _ k _ ij + _ ij _ k n^k + _ ij _ klm n^k ^lm + & & + ( _ i^k_klj + ^k_j_kli ) ( ^l - n^m _ ml + n^l ) + & & + _ i ( _ j - n^k _ kj + n_j ) - _ k ( ^k - n^l _ l^k+ n^k ) _ ij & ^ij = & ^jkl _ k ( n ( 1/2 _ l^i -_l^i ) ) - ^klm _ m ( n _ kl ) ^ij+ 2^jkm ^[ik ] _m ( n ) + & & + ( ^i ^j - ^k _ k ^ij ) n + n [ - 2 ^(ik ) _ k^j + ^ij - 2 ^[kl ] _ [ kl]^ij ] + & & + n^k _ k _ij + _ij _ k n^k + _ij _ klm n^k ^lm + & & + ( _i^k_klj + ^k_j_kli ) ( ^l - n^m _ ml + n^l ) in the reduced system the derivative terms are set to zero leading to the following equations of motion for and ._ ij = n [ 2 _ k(i ^k_j ) - _ij ] + 2 _ kl(i _ j)^k ( ^l - n^m _ m^l + n^l ) _ ij = n [ - 2 _ ik ^k_j + _ ij ] + 2 _kl(i _j)^k ( ^l - n^m _ m^l + n^l )

Abstract: the belinskii , khalatnikov and lifshitz conjecture posits that on approach to a space - like singularity in general relativity the dynamics are well approximated by ` ignoring spatial derivatives in favor of time derivatives . ' in we examined this idea from within a hamiltonian framework and provided a new formulation of the conjecture in terms of variables well suited to loop quantum gravity . we now present the details of the analytical part of that investigation . while our motivation came from quantum considerations , thanks to some of its new features , our formulation should be useful also for future analytical and numerical investigations within general relativity .