Eduardo. I. Guendelman

The string tensions can be dynamical in the modified measure formalism and appear as an additional dynamical degrees of freedom. These tensions may not be universal, instead, each string generates its own tension. We then consider a new bulk field that can couple to the strings, the tension scalar which changes locally the tension along the world sheet. In the case with two string tensions there is a braneworld solution which gives rise to an induced de Sitter space in the brane, avoiding swampland constraints of the standard string theory. Strings with different tension to ours can appear also as Dark Matter and since they share the same space and compactifications as visible matter, they should lead to Dark copies of the Standard Model.

We perform a late‐time cosmological study, we compare the performance of two Dirac–Born–Infeld (DBI) type k‐essence scalar field extensions of the model to the standard framework and a scenario using the Chevallier–Polarski–Linder (CPL) equation of state parametrization. We solve background dynamics numerically as functions of redshift and incorporate them into a Bayesian inference pipeline accelerated by machine learning. We use a Flax‐based surrogate emulator to replace repeated direct integrations of the ODE system, reducing computational cost. A hybrid scheme that combines stochastic variational inference (SVI) with No‐U‐Turn Hamiltonian Monte Carlo constrains cosmological parameters using the Pantheon+SH0ES Type Ia supernova sample, DESI BAO (DR2) data, and cosmic chronometer measurements without CMB‐based priors. In both DBI k‐essence formulations, present‐day dark energy equations of state are consistent with cosmic acceleration, indicating a ‐like regime with a modest redshift dependence. The model is marginally favored by conventional model selection measures such as , AIC, BIC, and DIC, which are based on goodness of fit and penalized. However, Bayesian predictive measures like WAIC and PSIS–LOO show no significant differences between , , and DBI k‐essence scenarios. All have similar model weights and out‐of‐sample predictive performance for the datasets. Thus, DBI k‐essence models mimic the success of the classic paradigm while allowing controlled, redshift‐dependent deviations from a strict cosmological constant that are consistent with present late‐time observations.

The string tensions can be dynamical, as we have studied in recent publications, for example in the case when we formulate string theories in the modified measure formalism. Then string and brane tensions appear, but as an additional dynamical degree of freedom. It can be seen however that these string or brane tensions may not be universal, but rather each string and each brane generates its own tension, which can have a different value for each string or brane. The case where there are strings that can have different spontaneously generated tensions has been considered in previous publication. To have a real dynamical string tension, we consider new background fields that can couple to the strings, the tension scalar which is capable of changing locally along the world sheet the value of the tension of the extended object. When all string tensions are equal, there is now an unbroken Target space scale invariance. By considering possible effective actions of gravity and strings as matter in D dimensions, we determine D by requiring that the effective action be, as the fundamental theory, target space scale invariant, which singles out [ ].

This is a comment on our previous paper (Guendelman in Eur Phys J C 85(6). https://doi.org/10.1140/epjc/s10052-025-14408-2). The string and brane tensions do not have to be put in by hand, they can be dynamically generated, as in the case when we formulate string and brane theories in the modified measure formalism. Then string tension appears, but as an integration constant. It can be seen however that these string tensions are not universal, but rather each string and each brane generates its own tension. To make the string tension fully dynamical, a bulk field (the tension field) is introduced. As we have seen in previous publications, world sheet conformal invariance in the case of two different species of strings with different tension can produce braneworlds. Now we add another crucial observation; dark matter to us may consist of matter made out of strings with different tensions because of decoupling of standard string interactions for strings with different tensions, although interactions mediated by the tension field can exist between strings of different tensions. Furthermore, Visible Matter strings and these “dark strings” share the same space time, and therefore the same compactifications, etc. Since the choice of compactification organizes the structure of the theory including its particle content, we expect therefore that the dark sector will consist of dark copies of the standard model and a visible Matter Dark Matter correspondence could be established.

In the modified measure formulation string or branes the tension appear as an additional dynamical degree of freedom . Furthermore in the presence of an additional background scalar field that couples to the strings and locally changes the tension, the tension field really dynamical and the theory has an intrinsic target space scale symmetry. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einsteins equations and the consistency of these two Einsteins equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einsteins equation . We review a case where two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation and leads to a well defined brane world solution. We review how the model avoid swampland constraints making treatments for Dark energy and inflation more realistic and how strings with a different tension appear as Dark Matter to us. Since the Dark strings and since the visible strings share the space time, including the same compactification, and since the compactification determines the particle content, we argue that the dark strings lead to Dark copies of the standard model. An argument that many copies will lead to 4D effective spacetime as a consequence of a target space scale symmetry restoration.

The string and brane tensions do not have to be put in by hand, they can be dynamically generated, as in the case when we formulate string and brane theories in the modified measure formalism. Here we take the measure as a metric independent lagrange multiplier that forces some lagrangian density to be zero. Then string tension appears, but as an integration constant. It can be seen however that these string tensions are not universal, but rather each string and each brane generates its own tension. To make the string tension fully dynamical, a bulk field (the tension field) is introduced. As we have seen in previous publications, world sheet conformal invariance in the case of two different species of strings with different tension can produce braneworlds, Swampland constraints may be avoided. Here we introduce braneworld scenarios without singularities of the string tensions, although they still can grow to very large values, this is done by demanding a certain periodicity in one light like coordinate. Now we add another crucial observation; Dark matter to us may consist of matter made out of strings with different tensions because of decoupling of standard string interactions for strings with different tensions, although interactions mediated by the tension field can exist between strings of different tensions.