Cosmological solutions and observational constraints on 5-dimensional braneworld cosmology with gravitating Nambu-Goto matching conditions
We investigate the cosmological implications of the recently constructed 5-dimensional braneworld cosmology with gravitating Nambu-Goto matching conditions. Inserting both matter and radiation sectors, we extract analytical cosmological solutions. Additionally, we use observational data from Type Ia Supernovae (SNIa) and Baryon Acoustic Oscillations (BAO), along with requirements of Big Bang Nucleosynthesis (BBN) and Cosmic Microwave Background (CMB) radiation, in order to impose constraints on the parameters of the model. We find that the scenario at hand is in good agreement with observations, and thus a small departure from the standard Randall-Sundrum scenario is allowed. However, the concordance CDM cosmology is still favored comparing to both standard braneworld model and the present scenario.
b,c]Emmanuel N. Saridakis
Cosmological solutions and observational constraints on 5-dimensional braneworld cosmology with gravitating Nambu-Goto matching conditions
Research Group of Geometry, Dynamical Systems and Cosmology, Department of Information and Communication Systems Engineering, University of the Aegean, Karlovassi 83200, Samos, Greece
Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece
Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4950, Valparaíso, Chile
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-3, Beijing 100049, China
Keywords: Braneworld models, dark energy, observational constraints, inflation
Israel matching conditions  are considered as the standard equations of motion of a classical codimension-1 defect which backreacts on the bulk dynamics. They are derived by focusing on the distributional part of the Einstein field equations (or some gravity modification) where the brane energy-momentum tensor, specified by a delta function, is included. An equivalent way to derive these equations is to take the variation of the brane-bulk action with respect to the induced metric, while the bulk equations of motion are derived as usually by varying the bulk action with respect to the bulk metric. However, a higher codimension defect carrying a generic energy-momentum tensor is known to be inconsistent with Einstein’s equations [2, 3, 4] (a brane with a pure tension is a special consistent case [5, 6, 7, 8, 9, 10, 11, 12, 13]). In  it was considered the idea that a more general theory like Einstein-Gauss-Bonnet gravity in six dimensions might remove the previous inconsistency, and the matching conditions of the theory for a generic energy-momentum tensor were derived. In  the consistency of the whole system of bulk field equations plus matching conditions was shown for an axially symmetric codimension-2 cosmological brane.
The spirit of the above proposal for consistency of the higher codimension defects is to include higher Lovelock densities [17, 18]. However, in dimensions, the highest such density is of order , and so, it is quite probable that branes with codimension higher than will still be inconsistent. Moreover, four-dimensions which represent effectively spacetime at certain length and energy scales do not allow generic codimension-2 or 3 defects. On the other hand, Israel matching conditions and their generalizations to higher codimensions do not accept the Nambu-Goto probe limit, which is the lowest order approximation of a test brane moving in curved background. Even the geodesic limit of the Israel matching conditions is questionable as a probe limit, since being the geodesic equation a kinematical fact it should be preserved independent of the gravitational theory (similarly to , ) or the codimension of the defect, which is not the case for these matching conditions [14, 21, 22, 23, 24, 25]. Moreover, even the non-geodesic probe limit of the standard equations of motion for various codimension defects in Lovelock gravity theories is not accepted, since this consists of higher order algebraic equations in the extrinsic curvature, and therefore, a multiplicity of probe solutions arise instead of a unique equation of motion at the probe level. In view of these observations a criticism to the standard matching conditions appeared in , where alternative matching conditions were proposed. These are the “gravitating Nambu-Goto matching conditions” which arise by the variation of the brane-bulk action with respect to the brane embedding fields, so that the gravitational back-reaction of the brane is taken into account. With these matching conditions a brane is always consistent for an arbitrary energy-momentum tensor and it also possesses the Nambu-Goto probe limit (the codimension-2 case was studied in , , while the codimension-1 in ). In  the application of these alternative matching conditions led to a new 5-dimensional braneworld cosmology which generalizes the conventional braneworld cosmology  in the sense that it contains an extra integration constant, and vanishing this constant gives back the standard braneworld cosmology.
In the current work we try to confront this cosmology with the current cosmological observational data (SNIa, BAO, BBN) in order to construct the corresponding probability contour-plots for the parameters of the theory. The paper is organized as follows: In section 2 we briefly present the alternative matching conditions and the basic features behind these, and we find in the cosmological framework the equation for the expansion rate including both the matter and radiation sectors. In section 3, which is the main part of the work, we impose the observational constraints on the parameters of the model. Finally, a summary of the obtained results is given in section 4 of conclusions.
2 5-dimensional braneworld with gravitating Nambu-Goto matching conditions
Our system is described by five-dimensional Einstein gravity coupled to a localized 3-brane source. The domain wall is assumed to be symmetric, it splits the spacetime into two parts and the two sides of are denoted by . The total brane-bulk action is
where is the (continuous) bulk metric tensor and is the induced metric on the brane with the unit normals pointing inwards ( are five-dimensional coordinate indices). The bulk coordinates are and the brane coordinates are ( are coordinate indices on the brane). The brane tension is and the matter Lagrangian of the brane is . The only matter content of the bulk is the cosmological constant and the higher dimensional mass scale is . The contribution on each side of the wall of the Gibbons-Hawking term is also necessary here as in the standard derivation of the matching conditions. is the trace of the extrinsic curvature (the covariant differentiation corresponds to ).
Varying (2.1) with respect to the bulk metric we get the bulk equations of motion
where is the bulk Einstein tensor. In this variation, beyond the basic terms proportional to which give (2.2), there appear, as usually, extra terms proportional to the second covariant derivatives which lead to a surface integral on the brane with terms proportional to . Adding the Gibbons-Hawking term, the normal derivatives of , i.e. terms of the form , are canceled, and considering as boundary condition for the variation of the bulk metric its vanishing on the brane (Dirichlet boundary condition for ) there is nothing left beyond the terms in equation (2.2). The Gibbons-Hawking term will again contribute in the following variation performed in order to obtain the brane equations of motion.
According to the standard method, the interaction of the brane with the bulk comes from the variation at the brane position of the action (2.1), which is equivalent to adding on the right-hand side of equation (2.2) the term , where , is the brane energy-momentum tensor and is the one-dimensional delta function with support on the defect. This approach leads to the standard Israel matching conditions. Here, we discuss an alternative approach where the interaction of the brane with bulk gravity is obtained by varying the total action (2.1) with respect to , the embedding fields of the brane position . The embedding fields are some functions and their variations are . While in the standard method the variation of the bulk metric at the brane position remains arbitrary, here the corresponding variation is induced by , i.e. . The result of this variation gives the codimension-1 gravitating Nambu-Goto matching conditions  (for a reminiscent variation see also )
where , and denotes covariant differentiation with respect to . These equations are supplemented with the bulk equations (2.2) which are defined limitingly on the brane, and therefore, additional equations have to be satisfied at the brane position beyond the matching conditions. Using these bulk equations the system of the above matching conditions (2.3), (2.4) is written equivalently as
where is the 3-dimensional Ricci scalar.
In order to search for cosmological solutions we consider the corresponding form for the bulk metric in the Gaussian-normal coordinates
where is a maximally symmetric 3-dimensional metric () characterized by its spatial curvature . The energy-momentum tensor on the brane (beyond that of the brane tension ) is assumed to be the one of perfect cosmic fluids with total energy density and total pressure .
The , bulk equations (2.2) at the position of the brane are
and a prime, a dot denote respectively , . The cosmic scale factor, lapse function and Hubble parameter arise as the restrictions on the brane of the functions and respectively. Other quantities also have their corresponding values when restricted on the brane, and since all the following equations will refer to the brane position, we will use the same symbols for the restricted quantities without confusion. Combining equations (2.8), (2.9) together with the matching condition (2.5) , we obtain the solution for
where is integration constant, and the Raychaudhuri equation for the brane cosmology
It is seen from (2.12) that for , the lower branch contains the Minkowski solution under the assumption of the Randall-Sundrum fine-tuning [31, 32]. We will not assume this condition in our analysis, so in the absence of matter our cosmology may have a de-Sitter vacuum. It is assumed that the quantity inside the square root of equation (2.12) is positive.
In  a single component perfect fluid was considered. Here, since we want to confront the model with real data, we will be more precise by assuming that the total energy density consists of the matter component with and the radiation component with , i.e. . Now, the integration process of (2.12) differs from that in . The variable
obeys the differential equation
Note that the Randall-Sundrum fine-tuning corresponds to the value . Using the conservation equation (2.6) in the standard form
we obtain the equation
Finally, changing to the variable
Each fluid component is conserved independently
so the solutions are
Therefore, equation (2.22) becomes a linear differential equation in terms of
with general solution
where is integration constant.
From the definition (2.21) we can find that
In this equation the sign index or has been used to denote
a new different bifurcation
from the previous branches. It is seen from (2.27) that the
sign is only
consistent with the lower branch, while the sign is
consistent with both branches.
The distinction, however, introduced by the sign index will be lost
in the expressions for the expansion rate and the acceleration parameter and
only the sign
will distinguish the two branches of solutions.
The expansion rate of the new cosmology arises by squaring equation (2.27) and is given by
where in (2.28) one can set . Redefining the integration constant as , the expansion rate can also be written as
where in (2.29) one cannot set since is in the denominator of the definition of c. This solution contains two integrations constants. The first constant is associated with the usual dark radiation term reflecting the non-vanishing bulk Weyl tensor. The second constant or c is the new feature that does not appear in the cosmology of the standard matching conditions  and signals new characteristics in the cosmic evolution. Setting in the branch we obtain the braneworld cosmology of the standard matching conditions (if there is no radiation we just set ). Of course, there are always the extra integration constants , of equations (2.24) which are adjusted by the today matter contents, while the today Hubble value is assumed to be given. The solution also contains three free parameters , , or , , . In  for a single dust perfect fluid, which approximates well at least the late-times behaviour, it was found analytically for values of extremely close to the Randall-Sundrum fine-tuning the position of the recent passage from a long deceleration era to the present accelerating epoch. Moreover, the age of the universe was estimated and the time variability of the dark energy equation of state was calculated.
3 Observational constraints
As we analyzed in detail above, the cosmological scenario at hand leads to the Friedmann equation (2.28), where the index corresponds to two branches of solutions. The Friedmann equation contains the following parameters: , , , and , along with , , . and are integration constants, is the fundamental 5D Planck mass, and the other two , are connected to the fundamental model parameters , and through the relations (2.17), (2.18). The identification of Newton’s constant in equation (2.28) as a combination of the model parameters will reduce the number of these parameters by one. Then, using we will define the various density parameters.
The scale factor for the branch with is bounded from above and we will not consider this case in detail. However, the branch with possesses the late-times asymptotic linearized regime (that is when , ) with a positive effective cosmological constant
Now, as usual in braneworld or other modified gravity models, from this late-times Friedmann equation, one reads the Newton’s constant. Since asymptotically the coefficients of in (3.1) are different, and , we associate Newton’s constant with
Finally, in order to complete the steps we rewrite (3.5) as
Note that this at late-times goes to which asymptotically goes to , i.e. to a simple cosmological constant.
So now, we can define the various density parameters straightforwardly as
Finally, assuming that the present scale factor is and using the redshift as the independent variable (), we can write the Friedmann equation (3.6) in the usual form, convenient to observational fittings
Here, , according to (3.7), is
In summary, Eq. (3.13) is the one we will fit, with , , , and as parameters (for simplicity we fix to their Planck + WP + highL + BAO best fit values, namely ). Concerning we include the direct probe from the Hubble Space Telescope (HST) observations of Cepheid variables with , that is we set it as a free parameter to fit the HST data.
The -term in (3.6) corresponds to dark radiation, so it is proportional to . This term, in particular , cannot be constrained efficiently by the low-redshift observations we are going to use in our analysis. However, since this dark radiation component was present at the time of Big Bang Nucleosynthesis (BBN) too, that is at redshift , we can use BBN arguments in order to constrain it. Specifically, the data impose an upper bound on the amount of total radiation (standard and exotic), which is expressed through the parameter of the effective neutrino species [34, 35, 36, 37]. Thus, in our case, this bound imposes a constraint on , namely
The recently released Planck results impose a quite tight constraint on the effective number of neutrino species : (95% C.L.) from the Planck+WP+highL+BAO data combination. Therefore, the 95% C.L. upper limit of is . This leads to a very tight constraint on the dark radiation component of the scenario at hand, namely (95% C.L.). Thus, we can safely neglect this term in the remaining analysis and the remaining parameters to be fitted are , , and .
As a starting analysis, let us fit the case where is set to its value that corresponds to the standard braneworld cosmological scenario , namely (which is exactly zero in the absence of radiation). Thus, in this case we have only three free parameters, namely , and . In Fig. 1 we provide the two-dimensional contour plots on (), using SnIa and SnIa+BAO data combinations. The details of the fitting procedure are presented in the Appendix. As we observe, when we use SnIa data only, the constraints on are relatively weak, namely at the confidence level. However, addition of the BAO data introduces an extra constraining power and the total constraint becomes tighter, namely ( C.L.) from SnIa+BAO data. Finally, as we describe in the Appendix, the efficiency of the fitting is quantified by , which for this case is .
Let us now proceed to the general case, that is considering as an additional free parameter. In the upper graph of Fig. 2 we present the contour plots of versus , while in the lower graph of Fig. 2 we depict the contour plots of versus . As we observe, the SnIa constraints on the parameter are much weaker than those of Fig. 1, due to the additional fitting variable. In particular, the 95% C.L. bound is (additionally note that the parameter space is now allowed by the SnIa data, exactly due to the presence of non-zero ). Concerning the SnIa data leads also to the relatively weak constraint (95% C.L.). However, for the combined SnIa with BAO data, the constraints become much tighter. At 95% confidence level they are and , while their best fit values are very close to 3 and 0 respectively. Finally, the corresponding is .
Let us now refer to the constraints of the Cosmic Microwave Background (CMB) radiation on the scenario at hand. One can use such high-redshift probes, in particular the distance information of CMB, the shift parameter and the acoustic scale , from WMAP9 or Planck data . Their definitions are and , where and denote the comoving distance to the decoupling epoch and the comoving sound horizon at , respectively. The current CMB data imply that at the Universe is dominated by matter, that is . If in our model we neglect , and terms, and we insert the present value of the critical density , then (3.6) becomes , where
Apparently, we deduce that if we want the term to be significantly smaller than , we need
Since , it is implied that if we desire to satisfy the CMB data we need .
The likelihood contours of the dimensionless quantity versus is shown in Fig. 3. We can then straightforwardly estimate that at 1 confidence level . Moreover, to give an estimate for the value of the brane tension , we use the relation , which leads to at 1 confidence level. That is , where is the current value of the energy density of the observed cosmological constant.
Finally, we close this subsection by examining the constraints on the model from the age of the universe. In general, the age of the universe is given by
where in the scenario at hand is given by equation (3.13). Thus, taking into account the constraints on the model parameters elaborated above, we can construct the contour plots of versus , which is presented in Fig. 4. We can then straightforwardly estimate the age in Gyr, finding at 1 confidence level (for the CDM model with the corresponding age is 13.5Gyr). We observe from equations (2.28) and (3.21) that larger values of the mass scale in the range found above correspond to larger values of the age of the universe. Thus, since larger ages are preferable, the most probable estimations for lie closer to the upper bound.
In summary, as we observe, the cosmological observations constrain and close to their Randall-Sundrum values, namely and ( in the case of radiation absence). However, note that the data allow for a departure from Randall-Sundrum scenario. In particular, although the present model has an additional parameter compared to Randall-Sundrum one, the corresponding is the same in two models, namely . This means that braneworld models with gravitating Nambu-Goto matching condition are in “equal” agreement with observations as the standard braneworld models.
Lastly, if we desire to compare the scenario at hand with the concordance paradigm of standard CDM cosmology, we can be based on the Akaike Information Criterion (AIC) 
where is the maximum likelihood achievable by the model (with the corresponding of the analysis) and the number of parameters of the model. Hence, we obtain the difference on the AIC between the standard CDM cosmology and our gravitating Nambu-Goto matching conditions model as
where is the difference of the number of parameters between the models. Thus, although in our model we obtain a similar to that of CDM cosmology (), the fact that we use two additional parameters gives . Thus, we deduce that CDM cosmology is more favored comparing to the scenario at hand, since the two extra parameters do not improve the late-times fitting behavior.
In this case, the full Friedmann equation (2.28) is
The branch is completely new comparing to the standard braneworld models since the scale factor is bounded from above for any value of . Therefore, contrary to the branch , here, there is no pure late-times linearization regime. However, expanding the expression (3.24), there is a term linear in , so Newton’s constant can also here be identified. More precisely it is , where … do not contain terms linear in , and . Therefore, associating with we have the identification
Going back to equation (3.24), we eliminate the parameter and we rewrite the expansion rate for as
This expression takes the standard form
In summary, Eq. (3.27) is the one we will fit, with , , , and as parameters (again for simplicity we fix to its (Planck+WP+highL+BAO) best fit values, namely ). Additionally, we include the direct probe from the Hubble Space Telescope (HST) observations of Cepheid variables with . Similarly to the previous subsection, we can safely neglect since it is negligible according to BBN analysis. Finally, instead of it proves more convenient to introduce the dimensionless quantity (3.18), namely , where is the present critical energy density of the Universe.
We use combined SnIa and BAO data to constrain , , and . In Fig. 5 we present the corresponding two-dimensional likelihood contours. Firstly, note that in this case is not theoretically restricted to values greater than 3 and in particular it is constrained in much smaller values, namely (95% C.L. upper limit). Additionally, note that since at late times acquires negative values, the constraint on is very close to zero, namely (95% C.L.). Due to the strong degeneracy between and , the constraints on are very different from those in the branch case, namely (95% C.L.). However, note that the minimal for this case is , that is much higher than that for the branch case, which means that the branch case is not favored by observations. This can be additionally seen by calculating the corresponding age of the universe, which is much smaller than the CDM value. However, although this branch is not favored by late-times observations, due to that at early times, it could still play an important role in the inflationary regime.
In this work we constrained an alternative 5-dimensional braneworld cosmology using observational data. The difference with the standard braneworld cosmology refers to the adaptation of alternative matching conditions introduced in  which generalize the conventional matching conditions. The reasons for this consideration are possible theoretical deficiencies of the standard junction conditions, namely the need for consistency of the various codimension defects and the existence of a meaningful equation of motion at the probe limit. Instead of varying the brane-bulk action with respect to the bulk metric at the brane position and derive the standard matching conditions, we vary with respect to the brane embedding fields in a way that takes into account the gravitational back-reaction of the brane onto the bulk.
The proposed gravitating Nambu-Goto matching conditions may be close to the correct direction of finding realistic matching conditions since they always have the Nambu-Goto probe limit (independently of the gravity theory, the dimensionality of spacetime or codimensionality of the brane), and moreover, with these matching conditions, defects of any codimension seem to be consistent for any (second order) gravity theory. Compared to the conventional 5-dimensional braneworld cosmology, the new one possesses an extra integration constant, which if set to zero reduces the new cosmology to the conventional braneworld one.
In the present work we extended the codimension-1 cosmology of  by allowing both a matter and a radiation sector in order to extract observational constraints on the involved model parameters. In particular, we used data from supernovae type Ia (SNIa) and Baryon Acoustic Oscillations (BAO), along with arguments from Big Bang Nucleosynthesis (BBN) in order to construct the corresponding probability contour-plots for the parameters of the theory.
Concerning the first () branch of cosmology, we found that the parameters and that quantify the deviation from the Randall-Sundrum scenario, are constrained very close to their RS values as expected. However, a departure from Randall-Sundrum scenario is still allowed, and moreover, the corresponding is the same for both models. This means that braneworld models with gravitating Nambu-Goto matching condition are in “equal” agreement with observations with standard braneworld cosmology. However, application of the AIC criterion shows that both standard braneworld cosmology, as well as the extended scenario of the present work, are less favored by the data if we compare them with the concordance CDM cosmology since the two extra parameters do not improve the fitting behavior. Furthermore, the obtained age of the universe is , which is an additional observational advantage of the model. Finally, concerning the fundamental mass scale , the current age estimations imply that the preferred values of lie well below the GeV scale.
Concerning the second () cosmological branch, which is completely new and with no correspondence in Randall-Sundrum scenario, we extracted the corresponding likelihood contours. Although this case is still compatible with observations, the corresponding minimal is much higher than that for the branch case, which means that this branch case is not favored by late-times observations. However, although this branch is not favored by late-times observations, due to that at early times, it could still play an important role in the inflationary regime.
In summary, cosmology with gravitating Nambu-Goto matching conditions offers an extension to the standard Randall-Sundrum scenario. Apart from interesting solutions, we see that it is in agreement with observations since the data allow for a small deviation from Randall-Sundrum cosmology. Therefore, it should be worthy to further study the cosmological implications of the model, such as the inflationary behavior and the late-times asymptotic features, since especially a successful inflationary regime is something that cannot be obtained in the framework of CDM cosmology.
The research of ENS is implemented within the framework of the Action “Supporting Postdoctoral Researchers” of the Operational Program “Education and Lifelong Learning” (Actions Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State. J.-Q. X. is supported by the National Youth Thousand Talents Program, the National Science Foundation of China under Grant No. 11422323, and the Strategic Priority Research Program âThe Emergence of Cosmological Structuresâ of the Chinese Academy of Sciences, Grant No. XDB09000000.
Appendix A Observational data and constraints
In this Appendix we review the main procedures of observational fittings used
in the present work, namely Type Ia Supernovae (SNIa) and Baryon Acoustic
a. Type Ia Supernovae constraints
We use the Union 2.1 compilation of SnIa data  in order to incorporate Supernovae type Ia constraints. This is a heterogeneous data set, which includes data from the Supernova Legacy Survey, the Essence survey and the Hubble-Space-Telescope observed distant supernovae.
The for this analysis is written as
where is the number of SNIa data points. In the above expression is the observed distance modulus, which is defined as the difference of the supernova apparent magnitude from its absolute one. Furthermore, are the errors in the observed distance moduli, which are assumed to be uncorrelated and Gaussian, arising from a variety of sources. If we introduce the usual (dimensionless) luminosity distance , calculated by