IIB matrix model and regularized big bang
Abstract
The large master field of the Lorentzian IIB matrix model can, in principle, give rise to a particular degenerate metric relevant to a regularized big bang. The length parameter of this degenerate metric is then calculated in terms of the IIBmatrixmodel length scale.
pacs:
98.80.Bp, 11.25.w, 11.25.YbarXiv:2009.06525 KA–TP–14–2020 (v4)
I Introduction
Einstein’s gravitational field equation Einstein1916 gives, in a cosmological context, the Friedmann–Lemaître–Robertson–Walker (FLRW) solution of a homogeneous and isotropic expanding universe with relativistic matter Friedmann19221924 ; Lemaitre1927 ; Robertson1935 ; Walker1937 . This solution has, however, a singularity with diverging energy density and curvature, the big bang singularity at cosmictime coordinate .
Recently, we have suggested another solution Klinkhamer2019rbb , which has an external length parameter . This solution has maximum values of the energy density and the Kretschmann curvature scalar proportional to and , respectively. In a way, the length parameter acts as a “regulator” of the big bang singularity and the new solution has been called the regularizedbigbang solution. This new solution replaces the Friedmann big bang curvature singularity at by a “spacetime defect” localized at . The spacetime defect is, in fact, described by a degenerate metric with a vanishing determinant at . The details of this new cosmological solution are discussed in Refs. Klinkhamer2020more ; KlinkhamerWang2019cosm ; KlinkhamerWang2020pert and further information on this particular type of spacetime defect appears in Refs. KlinkhamerSorba2014 ; Guenther2017 ; Klinkhamer2019JPCS .
Up till now, the length parameter of the degenerate metric is a mathematical artifact (regulator). But it is also possible that is actually a remnant of a new physics phase that replaces Einstein gravity. In App. B of Ref. Klinkhamer2020more and App. C of Ref. KlinkhamerWang2020pert , we have explicitly mentioned loop quantum gravity AshtekarSingh2011 and string theory GasperiniVeneziano2002 as possible candidates for the physics of this new phase. Especially interesting may be the nonperturbative formulation of string theory, which may hold some surprises in store for the nature of the new phase Witten2002 .
A particular formulation of nonperturbative typeIIB superstring theory (–theory) is given by the socalled IIB matrix model IKKT1997 ; Aokietalreview1999 . As this model involves only a finite number of matrices (traceless Hermitian matrices of size , where is taken to infinity), spacetime and gravity must emerge dynamically. Numerical simulations KimNishimuraTsuchiya2012 ; NishimuraTsuchiya2019 of the Lorentzian version of the IIB matrix model suggest, in fact, that a tendimensional classical spacetime emerges with three “large” spatial dimensions behaving differently from six “small” spatial dimensions. The previous literature IKKT1997 ; Aokietalreview1999 ; KimNishimuraTsuchiya2012 ; NishimuraTsuchiya2019 is, however, not entirely clear on from where precisely the spacetime points and metric come.
About a year ago, we suggested that, in the context of matrix models, the large master field Witten1979 may play a crucial role for the emergence of a classical spacetime. This suggestion was detailed in Ref. Klinkhamer2020master and several toymodel calculations were presented in two followup papers Klinkhamer2020points ; Klinkhamer2020metric .
We now pose the following question: does the master field of the Lorentzian IIB matrix model (assumed to be relevant for the physics of the Universe) give an emerging spacetime with a particular degenerate metric that corresponds to the regularizedbigbang solution of general relativity? At this moment, we cannot provide a definite answer, as we do not know the IIBmatrixmodel master field. But, awaiting the final result on the master field, we can already investigate what properties the master field would need to have in order to be able to produce, if at all possible, an emerging metric resembling the metric of the regularizedbigbang solution. (It is far from obvious that the IIBmatrixmodel expression for the emergent metric can give rise to such a type of metric.) The present paper is, therefore, solely exploratory in character.
Ii Background material
ii.1 Regularizedbigbang solution
In Sec. I, we have already mentioned the main properties of the regularizedbigbang solution in general relativity. Here, we will briefly recall the relevant expressions of this metric.
The new line element is given by Robertson1935 ; Walker1937 ; Klinkhamer2019rbb
(1a)  
(1b)  
(1c) 
where we have set and . The spacetime indices , run over and the spatial indices , over . Observe that the cosmictime coordinate covers the whole of the real line. The real function corresponds to the cosmic scale factor.
The metric from (1) is degenerate, having a vanishing determinant at , and describes a spacetime defect with characteristic length parameter . Further references on this type of spacetime defect have been given in Refs. Klinkhamer2019rbb ; Klinkhamer2020more . The similarities and differences of the RW and RWK metrics are also discussed in a recent review Klinkhamer2021APPBreview .
Assuming a homogeneous perfect fluid for the matter, with energy density and pressure , and inserting the metric (1) in the Einstein gravitational field equation (taking the appropriate limits Guenther2017 for ) produces a modified Friedmann equation for , which has a bouncetype solution with a nonsingular behavior of the energy density and curvature at . The matter of the homogeneous perfect fluid satisfies the standard energy conditions, for example the null energy condition .
Specifically, the modified spatially flat Friedmann equation, the energyconservation equation of the matter, and the equation of state are Klinkhamer2020more
(2a)  
(2b)  
(2c) 
where the overdot stands for differentiation with respect to and is Newton’s gravitational coupling constant. For relativistic matter (), the regular solution of (2) reads Friedmann19221924 ; Lemaitre1927 ; Robertson1935 ; Walker1937 ; Klinkhamer2019rbb
(3) 
with . The new solution (3) appears, in slightly different notation, as Eq. (3.7) in Ref. Klinkhamer2019rbb . The review Klinkhamer2021APPBreview elaborates on how the degenerate metric (1) with cosmic scale factor from (3) satisfies the Hawking–Penrose cosmological singularity theorems HawkingPenrose1970 . Further discussion of the resulting bouncing cosmology appears in Refs. KlinkhamerWang2019cosm ; KlinkhamerWang2020pert , but here we highlight only one result.
Consider a perturbative Ansatz around the “bounce” at ,
(4a)  
(4b) 
with the energydensity scale and real constants and , for . The modified Friedmann equation (2a) then gives the following parametric relation:
(5) 
where we have assumed and have set and in the definition of the Planck length [recall that, in general, we have ]. From the measured value of , we get ; see Ref. MisnerThorneWheeler2017 for further discussion. The relation (5) will be used in Sec. IV.
For comparison with later results, we give explicitly the RWK metric (a rank2 covariant tensor) corresponding to the line element (1),
(6) 
The inverse metric (a rank2 contravariant tensor) is simply given by the matrix inverse MisnerThorneWheeler2017 ,
(7) 
which, for , has a divergent component at .
ii.2 Emergent spacetime metric
The IIB matrix model is extremely simple to formulate, having a finite number of matrices, but extremely hard to evaluate and interpret. More specifically, the model has a finite number of traceless Hermitian matrices ( is taken to infinity later). Details of the IIB matrix model are given in the original papers IKKT1997 ; Aokietalreview1999 and have been reviewed in Ref. Klinkhamer2020master . Here, we only recall what is needed for the further discussion.
Adapting Eq. (4.16) of Ref. Aokietalreview1999 to our masterfield approach, we have obtained the following expression for the emergent inverse metric Klinkhamer2020master :
(8) 
with spacetime dimension for the original matrix model and continuous spacetime coordinates . These spacetime coordinates have the dimension of length, which traces back to the IIBmatrixmodel length scale that has been introduced in the path integral Klinkhamer2020master . The average in the integrand of (8) will be discussed shortly, after some other explanations have been given.
We refer to Refs. Klinkhamer2020master ; Klinkhamer2020points for the details on how the discrete spacetime points , with index , are extracted from the bosonic master field . This bosonic master field corresponds to a set of ten traceless Hermitian matrices for , with positive integers and . The limit carries along the limit , provided stays constant or increases (the role of will be explained below).
The quantities entering the integral (8) are the density function
(9) 
for the emergent spacetime points as obtained in Refs. Klinkhamer2020master ; Klinkhamer2020points and the dimensionless density correlation function defined by
(10) 
In (8), there is also a localized symmetric real function , which appears in the effective action Aokietalreview1999 ; Klinkhamer2020master of a lowenergy scalar degree of freedom hopping over the discrete spacetime points ,
(11) 
where is the field value at the point (the scalar degree of freedom arises from a perturbation of the master field and has the dimension of length; see App. A of Ref. Klinkhamer2020master for a toymodel calculation). As this function , the inverse metric from (8) is manifestly dimensionless. The metric is obtained as the matrix inverse of . has the dimension of
The extraction procedure Klinkhamer2020master ; Klinkhamer2020points of the discrete spacetime points relies on blocks positioned adjacently along the diagonal of the matrices of the bosonic master field (there are blocks on each diagonal). Very briefly, the coordinates of the discrete spacetime points are obtained as follows: the th block on the diagonal of the single matrix (where is a fixed index) gives a single coordinate as the average of the real eigenvalues of this particular block. The average entering (8) and (10) then corresponds to averaging over different block sizes and different block positions along the diagonal in the master field. The details of this averaging procedure still need to be clarified, but this does not affect the present discussion.
A few heuristic remarks may help to clarify expression (8) for the emergent inverse metric. In the standard continuum theory [i.e., a scalar field propagating over a given continuous spacetime manifold with metric ], two nearby points and have approximately equal field values, , and two distant points and generically have different field values, . The logic is inverted for our discussion. Two approximately equal field values, , still have a relatively small action (11) if and inserting in (8) gives a “large” value for the inverse metric and, hence, a “small” value for the metric , meaning that the spacetime points and are close (in units of ). Similarly, two very different field values and have a relatively small action (11) if and inserting in (8) gives a “small” value for the inverse metric and, hence, a “large” value for the metric , meaning that the spacetime points and are separated by a large distance (in units of ).
In the following, we will focuss on the four “large” spacetime dimensions KimNishimuraTsuchiya2012 ; NishimuraTsuchiya2019 and we have, for the emergent inverse metric,
(12a)  
(12b) 
with an effective spacetime dimension and the abbreviated notation . Perhaps it is even not necessary to do this additional averaging of in the integrand of (12a), as that is already taken care off by the limit Klinkhamer2020metric .
In Ref. Klinkhamer2020metric , we have rewritten the integral (12a) somewhat by using the integration variables and introducing new functions and . The resulting integral and the required definitions are
(13a)  
(13b)  
(13c) 
where the new function has a more complicated dependence on and than the combination , but the function is still symmetric, . The advantage of using (13a) is that the dependence in the integrand has now been insolated in only two functions, and .
For later use, we recall that the action of the tendimensional Lorentzian IIB matrix model IKKT1997 ; Aokietalreview1999 ; KimNishimuraTsuchiya2012 ; NishimuraTsuchiya2019 contains coupling constants , for indices
(14) 
We emphasize that the above numbers are only coupling constants and not yet a metric.
The purpose of the present paper is to investigate the integral (13a). It is not at all obvious that a Lorentzian inverse metric could appear with the required singular behavior. Indeed, we want to determine what would be required of the unknown functions , , and [which trace back to the IIBmatrixmodel master field], so that the integral (13a) gives the RWK inversemetric (7).
Iii Emergent degenerate metric
iii.1 Basic idea
By choosing an appropriate length unit, we set the IIBmatrixmodel length scale to unity, . In this way, the coordinates of the discrete emerging spacetime points are effectively dimensionless, and the same holds for the continuous spacetime coordinates used in Sec. II.2. Moreover, we write, in a cosmological context, these continuous spacetime coordinates as follows:
(15a)  
(15b) 
where is interpreted as the cosmictime coordinate and is set to unity by an appropriate choice of the time unit. The cosmictime coordinate is also effectively dimensionless.
In order to obtain an inverse metric with a possibly divergent component at , the convergence properties of the integral in (13a) need to be relaxed. Instead of the factor in as used by Ref. Klinkhamer2020metric for the standard spatially flat Robertson–Walker inverse metric, we consider the following structure of the function entering (13):
(16) 
Focussing on the integral and neglecting other contributions, we then have
(17a)  
(17b) 
where the first integral diverges linearly as but not the second.
Next, we must obtain and we use, for that, the following Ansatz:
(18) 
where is identified with the cosmictime coordinate and where, later, we set . Note that the above function , with equal and , is symmetric in its arguments and , which explains the appearance of the second term proportional to .
From (13) with and the Ansätze (16) and (18), we find that the integrals with can be done analytically. The integrals with are more complicated but can be dealt with after a Taylor expansion with respect to . The following structure is obtained:
(19a)  
(19b) 
Further work is needed to get a independent term in exactly equal to unity and the Lorentzian signature. In a first reading, it is possible to skip the technical details and move forward to Sec. III.4.
iii.2 Core structure
With the basic idea of the previous subsection [namely, a mild cutoff on the integral of (13) at values of order ], we have not yet obtained the core structure of the desired inverse metric (7). For that, we need an extended Ansatz with additional freedom carried by four real parameters . Remark that “core structure” refers to the inner structure of the spacetime defect Klinkhamer2019rbb , which, in this case, concerns the time coordinate and corresponds to a divergent component.
Specifically, we take the following Ansatz functions:
(20a)  
(20b)  
(20c)  
(20d) 
where we set, as before, and . One of the constants or in (20b) is superfluous, but we keep them both in order to ease the comparison with the previous calculation of Ref. Klinkhamer2020metric . The Ansatz involves, in addition, the coupling constants from the Lorentzian IIB matrix model reduced to dimensions, as given by (14). For a different way of obtaining a Lorentzian signature in the emergent inverse metric, see App. B of Ref. Klinkhamer2020master .
Inserting the Ansatz functions (20) into the emergentinversemetric expression (13), we can perform all integrals analytically, except for the integral involving the term. For that integral, we make a Taylor expansion in and then integrate analytically the resulting Taylor coefficients. As explained in Ref. Klinkhamer2020metric , we set
(21a)  
(21b) 
and obtain the following result ( is the spatial index):
(22a)  
(22b) 
with all other components vanishing by symmetry [the integrand of (13) then has a single factor , , or ]. The coefficients in (22) are functions of the four real Ansatz parameters .
In order to simplify the discussion, we immediately fix
(23) 
so that we only need to determine the appropriate values of the parameters and . In fact, the coefficient now only depends on the parameter , as is absent and and have been fixed to the numerical values (23). From the requirement
(24) 
where the tilde indicates the use of (23), we obtain a seventhorder algebraic equation for , which has two positive real roots. The analytic expressions for these two roots are rather cumbersome and we will just give their numerical values,
(25a)  
(25b) 
For definiteness, we take the first root from (25) and set
(26) 
Having found a suitable value for , we turn to the resulting coefficient of the inversemetric component . From the requirement
(27) 
where the tilde indicates the use of (23) and (26), we obtain a linear equation for and find the following solution:
(28) 
To summarize, we have, from the Ansatz functions (20) and the parameters
(29) 
the following result for the emergent inverse metric as given by the expression (13):
(30) 
where the numerical value of is of order (the actual numerical value will be given shortly).
Comparing to the generalrelativity inverse metric (7), we interpret the first two nontrivial terms of from (30) as follows:
(31a)  
(31b) 
where is the length scale of the IIB matrix model that we have previously set to unity. With the parameter values (29), we have
(32) 
but different numerical values are obtained if, for example, the value is changed away from the value or if the root is chosen instead of . The general parametric behavior of the coefficient in follows by adapting the elementary integral for in Sec. III.1 and gives
(33) 
for the particular Ansatz (20).
iii.3 First approximation
In the previous subsection, we have shown that, in principle, the emergent inversemetric expression (13) can give the core structure of the RWK inverse metric (7), with an explicit numerical value of the classicalgravity length parameter in units of the IIBmatrixmodel length scale . See, in particular, the results (31), (32), and (33).
We now want to check that the higher order terms in of (31a) can be made to vanish. For that, we will use, instead of (20c), a nontrivial Ansatz of the function. Specifically, we take
(34) 
with real parameters and an explicit exponential factor to guarantee the convergence of the integral (the terms will, for this reason, not modify the coefficient of the term in ). Keeping the parameter equal to the numerical value from (26) but allowing for a change in the numerical value of , we find that the coefficients of (22) have the following dependence:
(35a)  
(35b)  
(35c) 
where the overbar indicates the use of the numerical values (23) and (26).
Demanding
(36) 
gives three algebraic equations for the three parameters with the following solution:
(37a)  
(37b)  
(37c) 
which all three are functions of the free parameter . [Remark that is not a solution of the conditions (36).] The corresponding inverse metric reads
(38a)  
(38b)  
(38c) 
which is a significant improvement compared to the corestructure result (30). The result (38) corresponds, in fact, to a first approximation of the desired inverse metric valid to order .
We can invert the map (38c) and obtain the required input value for a desired value of ,
(39) 
In this way, we can get any Taylor coefficient in the component from (38) by choosing an appropriate value of the Ansatz parameter .
From (38a), we obtain by matrix inversion the diagonal metric which has the following component:
(40) 
It is already clear that this metric is degenerate, with a vanishing determinant at , but we postpone further discussion of this point to the next subsection.
iii.4 Conjectured final result
As indicated on the lefthand side of (38a), we consider that result to be a first approximation of the RWK inverse metric, as derived from the IIBmatrixmodel master field under the assumptions stated. Better approximations, with more and more Taylor coefficients for and more and more terms vanishing in , will follow from higher orders in the Ansatz function from (34) and possible further extensions of the Ansatz functions and . This procedure has been tested in Ref. Klinkhamer2020metric for the standard spatially flat Robertson–Walker inverse metric.
The final result for the emerged inverse metric is expected to have the following structure (in units with ):
(41) 
where the question mark indicates that, strictly speaking, this is a conjectured result. The in (41) are real dimensionless coefficients that result from the requirement that terms, for , vanish in . The emerged metric is given by the matrix inverse of (41),
(42) 
which has, for , a vanishing determinant at . In short, the emergent metric (42), obtained from the expression (13) with appropriate Ansatz functions and parameters, is degenerate.
The emergent metric (42) has indeed the structure of the RWK metric (6), with the following effective parameters:
(43a)  
(43b) 
where the IIBmatrixmodel length scale has been restored and where we omit the question marks as we have explicit results for the coefficients shown. Indeed, the leading coefficients are given by from (32) and from (38c), for the particular Ansatz functions (20a), (20b), and (34) and Ansatz parameters (23) and (26). If the Ansatz parameter in (38c) is chosen appropriately, we get in the square of the cosmic scale factor (43b), so that the emerged classical spacetime corresponds to the spacetime of a nonsingular cosmic bounce at , as obtained in Refs. Klinkhamer2019rbb ; Klinkhamer2020more from Einstein’s gravitational field equation. The proper cosmological interpretation of the emerged classical spacetime will be discussed further in Sec. IV.
Iv Conclusion
In the present article, we have started an exploratory investigation of how a new physics phase can give an emerging classical spacetime with an effective metric where the bigbang singularity has been tamed Klinkhamer2019rbb .
In order to be explicit, we have used the IIB matrix model IKKT1997 ; Aokietalreview1999 , which has been suggested as a nonperturbative definition of typeIIB superstring theory. If we interpret the numerical results KimNishimuraTsuchiya2012 ; NishimuraTsuchiya2019 from the Lorentzian IIB matrix model as corresponding to an approximation of the genuine master field Witten1979 , then it appears that spacetime points emerge with three “large” spatial dimensions and six “small” spatial dimensions. But the numerical simulations are still far removed from providing results on the required density and correlation functions that build the inverse metric Aokietalreview1999 ; Klinkhamer2020master .
For the moment, we have adopted a leapfrogging strategy by jumping over the actual analytic or numeric evaluation of the IIBmatrixmodel master field and by simply assuming certain types of behavior of the density and correlation functions that enter the inversemetric expression (8). The explicit goal of the present article is to establish what type of functions are required in (8) to get, if at all possible, an RWK inverse metric with the behavior shown in (7). [Note that, in principle, the origin of the expression (8) need not be the IIB matrix model but can be an entirely different theory, as long as the emerging inverse metric is given by a multiple integral with the same basic structure.]
For the integral (8), we have indeed been able to find suitable functions (these functions are, most likely, not unique), which give an emerging classical spacetime with an effective metric where the bigbang singularity has been tamed. In fact, the bigbang singularity is effectively regularized by a nonzero length parameter that is now calculated in terms of the IIBmatrixmodel length scale ; see the last paragraph of Sec. III.4. One important lesson, from the comparison with our previous calculation Klinkhamer2020metric of the Minkowski and Robertson–Walker metrics, appears to be that the relevant correlation functions must have longrange tails in the time direction, in order to get a divergent behavior of , as explained in Sec. III.1.
Remark that we have not yet obtained the effective (Einstein?) gravitational field equation and the corresponding solution of the metric. Instead, we have used a general constructive expression for the inverse metric, as given by (13) after some redefinitions. The further consistency of the emerging field theories may then restrict the values of some of the parameters entering our explicit Ansatz functions (20a), (20b), and (34), fixing, for example, the values of and , or even demanding different functional forms of the functions , , and .
Expanding on the previous paragraph, we observe that the IIB matrix model not only produces a classical spacetime but also its matter content Aokietalreview1999 . Now, the IIB matrix model in the formulation of Ref. Klinkhamer2020master has a single length scale , so that, for the cosmological quantities (4) near the bounce at , we expect an energydensity scale . If, moreover, general covariance Aokietalreview1999 and the Einstein gravitational field equation are recovered, we have from the relation (5) with the following parametric relation:
(44) 
where corresponds to (using units to set and to unity) and where the question mark indicates that this is a conjectured result. If correct, the emergent Planck length would, not surprisingly, be of the same order as the IIBmatrixmodel length scale . Reading (44) from the right to the left and inserting the experimental values for , , and on the lefthand side, we would also have an estimate for the actual value of the unknown IIBmatrixmodel length scale ,
(45) 
where the “experimental” numerical value for the Planck length was already given a few lines below (5).
The cosmological interpretation of the emerged classical spacetime is perhaps as follows. The new physics phase is assumed to be described by the IIB matrix model and the corresponding large master field gives rise to the points and the metric of a classical spacetime. If the master field has an appropriate structure, the emerged metric has a tamed big bang, with a metric similar to the RWK metric of general relativity Klinkhamer2019rbb , but now having an effective length parameter proportional to the IIBmatrixmodel length scale . In fact, one possible interpretation is that the new physics phase has produced a universeantiuniverse pair BoyleFinnTurok2018 , that is, a “universe” for and an “antiuniverse” for .
As a final comment on our main result from (43a) and the conjectured result (44), we recall that we have used a IIBmatrixmodel length scale that was introduced directly into the path integral Klinkhamer2020master . But a more subtle origin of the length scale is certainly not excluded. One example of such an origin would be, in the emerging massless relativistic quantum field theory from the matrix model, the appearance of a length scale by the phenomenon of dimensional transmutation ColemanWeinberg1973 . In any case, assuming the IIB matrix model to be relevant for physics, progress on fundamental questions such as the origin of the length scale or the birth of the Universe will only happen if more is known about the IIBmatrixmodel master field.
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